Photonic crystal (PhC) surface emitting lasers utilizing a 2-D distributed feedback (DFB) mechanism has a considerable amount of publication during the past few years [1-4]. The PhC lasers have such excellent advantages to attract the people’s attention including controlling the specific lasing modes such as longitudinal and transverse modes, lasing over the large area, and narrow divergence beam. Besides, the calculation of the photonic band-gap and the distribution of electric or magnetic field became more and more important. In the past few years, there were many theoretical calculations and methods have been developed, such as 2-D plane wave expansion method (PWEM) [2, 5], finite difference time domain (FDTD) [6, 7], transfer matrix method, and multiple scattering method, etc. However, there is no any detail lasing characteristic of PhC band-edge modes including the diagram of angular-solved μ-PL system, and the feature of high order lasing modes. In the discussion, we focused on the lasing behavior of PhC band-edge modes lasers in GaN-based 2-D PCSELs with AlN/GaN distributed Bragg reflectors. Each of PhC band-edge modes exhibits a different type of wave coupling mechanism according to the Bragg diffraction mechanism. In this chapter, we introduced the fundamental and higher order Bragg diffraction in section 4.1. According to Bragg diffraction mechanism and coupling wave theory, we can expect the fundamental and high order PhC lasing modes have specific lasing emission characteristics.
4-1 Bragg diffraction theory
First order Bragg diffraction in 2-D triangular lattice PhC Figure 4. 1
[8,9]
(a) shows a band diagram of PhC with triangular lattice. The points (A), (B), (C), (D), (E), and (F) present the different lasing modes including Γ1, K2, M1, Γ2, K2, and M2,
respectively. Each of the different PhC band-edge lasing modes represents the PhC nanostructure can control the light propagated in different lasing wavelength and band-edge region. Figure 4. 1(b) shows a schematic diagram in a reciprocal space. The reciprocal space of the PhC nanostructure is a space transferred by hexagonal photonic crystal nanostructure in real space. The K1 and K2 are the Bragg vectors with the same magnitude, |K|=2π/a, where a is the lattice constant of the photonic crystal. Considered the TE modes in the 2-D photonic crystal nanostructure, the diffracted light
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wave from the PhC structure must satisfy the Bragg’s law and energy conservation:
... where kd is a xy-plane wave vector of diffracted light wave; ki is a xy-plane wave vector of incident light wave; q1,2 is order of coupling; ωd is the frequency of diffracted light wave, and ωi is the frequency of incident light wave. Eq. (4. 1) represents the phase-matching condition (or momentum conservation), and Eq. (4. 2) represents the constant-frequency condition (or energy conservation).
When both of equations are satisfied, the lasing behavior would be observed.
(D)
Figure 4. 1 (a) The band diagram of photonic crystal with triangular lattice; (b) The schematic diagram of photonic crystal with triangular lattice in reciprocal space.
Of course, it is expected the lasing behavior would occur at specific points on the Brillouin-zone boundary including Γ, M, and K and these PhC band-edge lasing modes would split and cross. At these PhC lasing band-edge modes, waves propagating in different directions would be coupled and increased the mode density (or density of state, DOS). It is particularly interesting that each of these band-edge modes exhibits a different type of wave coupling routes. For example, as shown in Figure 4. 1(c), the coupling at point (C) only involves two waves, propagating in the forward and backward directions. This coupling is similar to that of a conventional DFB laser. Both of them show the similar coupling mechanism but different lasing behavior according to the different structure. However, there can be six equivalent Γ-M directions in the structure; that is, the
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cavity can exist independently in each of the three different directions to form three independent lasers. Point (B) has a unique coupling characteristic which is different resonance mechanism compared with the conventional DFB lasers. The coupling waves propagating in three different directions are shown in Figure 4. 2(b). This figure means that the cavity is a triangular shape. On the other hand, the point (B) can also be six Γ-K directions in the structure. Therefore, two different lasing cavities in different Γ-K directions coexist independently. At point (A) in Figure 4. 2(a), the coupling waves in in-plane are including six directions 0°, 60°, 120°, -60°, -120°, and 180. The coupled light can emit perpendicular from the sample surface according to satisfied first order Bragg diffraction, as shown in Figure 4. 3. Therefore, the PhC devices can function as a surface emitting lasers. kd indicate the incident and diffracted light wave.
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ki K1
kd=ki+K1
Figure 4. 3 The wave vector diagram at point (A) in vertical direction.
Higher order Bragg diffraction in 2-D PhC with triangular lattice
Figure 4. 4(a) and Figure 4. 4(b) show the in-plane and vertical diffraction at point (D). In this case, the light wave is diffracted in five Γ-K directions and in the vertical direction similar to point (A) and (ki+q1K1+q2K2) which can reach to the six Γ’ points. Figure 4. 4(c) shows the wave vector diagram of one Γ’ point in K space where the light wave is diffracted to an oblique direction. The light wave would be also diffracted to a bottom oblique direction.
Figure 4. 5(a) and (b) show the in-plane and vertical diffraction at point (E). In this case, the light wave is diffracted in three Γ-K directions and (ki+q1K1+q2K2) which can reach to the three K’
points. Figure 4. 5(b) shows the wave-vector diagram of one K’ point where the light wave is diffracted to an angle tilt 30˚ normally from the sample surface. Therefore, we expect the lasing behavior of K2 mode that would emit at this specific angle.
Figure 4. 6(a) and (b) show the in-plane and vertical diffraction at point (F). In this case, the light wave is diffracted in two different Γ-M directions and (ki+q1K1+q2K2) which can reach to the three M’ points. Figure 4. 6(b) shows the wave-vector diagram of one M’ point where the light wave is diffracted into three independent angles tilted of about 19.47˚, 35.26˚, and 61.87˚ normally from the sample surface, respectively. Since we collected PL spectrum on one detected plane, these diffraction angles would be happened on different detected planes. In our experiment, we could only detect one diffraction angle at one time limited by the detector when the PhC effect is occurred.
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Figure 4. 4 Wave vector diagram of (a) in-plane and (b) vertical direction at point (D); (c) wave vector diagram showing diffraction to an oblique direction at point (D).
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Kd=ki+2K1 ki
kd=ki+2K1-2K2 kd=ki+2K1-K2 Kd=ki+K1
kd=ki+K1-K2
(a)
30˚
ki kd
(b)
Figure 4. 5 Wave vector diagram of (a) in-plane and (b) vertical direction at point (E) (or K2 mode); ki and kd indicate incident and diffracted light wave.
kd=ki+2K1-K2 kd=ki+K1
kd=ki+K1-K2 ki
kd=ki+3K1
(a)
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19.47˚
35.26˚ 61.87˚
ki kd
kd kd
(b)
Figure 4. 6 Wave vector diagram of (a) in-plane and (b) vertical direction at point (F) (or M3 mode); ki and kd indicate incident and diffracted light wave.
Diffraction pattern of 2-D PhC
In the previous section, we show the Bragg diffraction mechanism causing different PhC lasing resonance routes including Γ mode and high order modes in K space. Furthermore, in this section, we will discuss about the “continuous” Bragg diffraction extraction mechanism causing different diffraction lines in the diagram of angular-resolved μ-PL system. Each direction of emission (or extraction) is associated with a given in-plane wave vector, k//, or effective index neff = (k//)/k0, where k0 is the wave vector of light in vacuum [ 11 ]. Therefore, the diagram by angular-resolved μ-PL system can be discussed and classified into four different parts by neff as shown in Figure 4.7(a). The parts of the spectrum with effective index neff < 1 are extracted directly.
These propagate in all directions in air and represent only 10% of the whole radiated power. The parts with 1 < neff < 1.7 are delocalized modes. These are produced by evanescent waves emitted by the dipole, and their contribution to the total radiated power is about 40%. The peaks in the dipole power spectrum with 1.7 < neff < 2.5 are associated with guided modes and also induced by the dipole emission of evanescent waves. They carry slightly more than 45% of the total emission. The components with neff > 2.5 are purely evanescent and do not contribute in the radiative intensity.
[10]
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(a) (b)
Figure 4. 7 (a)Normalized far-field PL spectrum of the PhC-assisted QD structure; (b) Reciprocal lattice associated with the 2-D PhC and origin of extracted guided modes. The blue and gray circles indicate the light cone and the trace of points with identical k//, respectively. The gray hexagon is the first Brillouin zone boundary.
For each reduced frequency, the measurement results covers all possible k// values in the light cone delimited by the air light line and seen clearly in this plot because no light can emit below the light line. Another characteristic line arises from the presence of the cutoff frequency for any given guided mode. This occurs when k// reaches the sapphire line, defined by k// = 1.7k0. For any given frequency, there is a discrete number of guided modes carried by the planar cavity, with 1.7 < neff <
2.5. The lowest order mode has neff < 2.5. This mode almost perfectly follows the GaN line, defined by k// = 2.5k0. The number of guided modes as measured with this sample is in accordance with the simulation photonic band diagram. To explain all of the effects caused by a 2-D PhC particular in the effects related to polarization, the field, associated with a guided modes for example, should be described as a Bloch mode: E(r) = ΣG EG * exp [i (k// + G) •r], where EG is the electric field component corresponding to harmonic G, and k// is the in-plane wave vector of the Bloch mode.
With our PhC structure, the reciprocal lattice (RL) in K space is a 2-D triangular lattice rotated by 30° with respect to the direct lattice (DL) in real space and RL vectors can be written as: G = ha1*
+ ka2*
, where h and k are integers, and a1*
and a2*
are the two RL basis vectors. Harmonics of the Bloch mode are extracted if their in-plane wave vectors are within the light cone: |k// + G| < k0. The most striking feature observed in Figure 4. 7(a) is the detection of the radiative components of
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guided modes. The sets of lines labeled 2a and 2b are induced by the radiative harmonics of the TE-polarized guided modes propagating in the Γ-M direction with in-plane wave vectors k// + G10
and k// + G-10 (shown in Figure 4. 7(b)), where shown only a radiative harmonic associated with set 2b, for clarity; 2a is obtained by symmetry. The sets of lines labeled 3a and 3b are formed by the combination of two harmonics (as shown in Figure 4. 8 for a line associated with set 3a). These radiative harmonics are not associated with guided modes propagating in the Γ-M direction but in directions about ±60°. The measurement of these components constitutes direct evidence of 2-D PhC-assisted light extraction [12].
Figure 4. 8 2-D band structure of a PhC in the Γ-M direction; (b) Corresponding band structure in a multimode waveguide: mode m gives rise to two PhC bands, Am and Bm.
4-2 Couple wave theory [13,14]
Distributed feedback lasers do not utilize the conventional cavity mirrors, but provide feedback via backward Bragg scattering from periodic perturbations of the refractive index or the gain of the medium. Distributed feedback structures are compact and provide a high degree of spectral selection. In this section, we are focus more on the electromagnetic aspects of light wave propagation, particularly for our photonic structures. We revolve around couple-wave theory to approximately solve the complex equations, which would be addressed numerically.
In order to use these approaches, we generally know at least that some of eigenmodes of a relatively simple waveguide configuration. The trick is to express the solution to some perturbed or more complex configuration in terms of these original basis set of eigenmodes. Then we can get general form of any dimension couple-wave equation.
1-D Couple Wave Theory
To get started, we recall the fundamental wave equation to help us understand it. In a
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homogeneous, source-free and lossless medium, any time dependent harmonic electric field satisfy the vector wave equation
∇2E+k02n2E =0
(4. 3) where the time dependence of the electric field is assumed to be ejwt , n is the refractive index and k0
is the free space propagation constant. And the electric field must satisfy the homogeneous wave equation such that:
2 0
2
2 E+ Ek =
δz
δ (4. 4)
Consider a multi-dielectric stack in which periodic corrugations are formed along one boundary as illustrated in Figure 4. 9.
Figure 4. 9 General multi-dielectric layers show the perturbation of refractive index and amplitude gain. Z1(x) and Z2(x) are two corrugated functions.
The material complex permittivity in each layer is denoted as εj while g and Λ are the height and the period of corrugation, respectively. With corrugations extending along the longitudinal direction, the wave propagation constant, k(z), could be written as
k2(z)=w2µε' (4. 5) where w is the angular frequency and ε’ is the complex permittivity. When the radiation frequency is sufficiently close to the resonance frequency, Eq. (4. 5) becomes
78 where n(z) and α(z) are the refractive index and the amplitude gain coefficient, respectively. Within the grating region dx≤ x≤dx+g, perturbation is considered so the refractive index and gain coefficient can be expressed in a Fourier form as
n(z)=n0+∆ncos(2β0z+Ω) (4. 7) and
α(z)=α0+∆αcos(2β0z+Ω+θ) (4. 8) Here, n0 and α0 are the steady-state values of the refractive index and amplitude gain, respectively. ∆n and ∆α are the amplitude perturbation terms, β is the propagation constant 0 and Ω is the non-zero residue phase at the z-axis origin. In the Eq. (4. 8), θ express the relative phase difference between perturbations of the refractive index and amplitude gain. Assume there is an incident plane wave entering the periodic and lossless waveguide at an angle of Φ as shown in Figure 4. 10.
Figure 4. 10 A simple model used to explain Bragg conditions in a periodic waveguide.
At each periodic interval of Λ, the incident wave will experience the same degree of refractive index change so that the incident wave will be reflected in the same direction. For a waveguide that consists of N periodic corrugations, there will be N reflected wavelets. In order that any two reflected wavelets add up in phase or interfere constructively, the phase difference between
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the reflected wavelets must be a multiple of 2π. In other words,
β0(AB+BC)=β0(2ΛsinΦ)=2mπ (4. 9) where m is an integer. If the incident wave is now approaching more or less at a right angle to the wavefront (i.e.
π2
≈
Φ ), Eq. (4. 9) becomes
2β0Λ=2mπ (4. 10) This is known as the Bragg condition and β becomes the Bragg propagation constant. The 0
integer m shown in the above equation defines the order of Bragg diffraction. Unless otherwise stated, first-order Bragg resonance (m =1) is assumed. Since a laser forms a resonant cavity, the Bragg condition must be satisfied [13].Rearranging Eq. (4. 10) gives
= Λ
11), it is clear that the Bragg propagation constant is related to the grating period. By altering the grating period, the Bragg wavelength can be shifted according to the specific application.
Using small signal analysis, the perturbations of the refractive index and gain are always smaller than their average values, i.e.
n0
n<<
∆ , ∆α <<α0 (4. 12) Substituting Eq. (4. 7) and Eq. (4. 8) into Eq. (4.6) using the above assumption, generates
k2(z)=k02n02 + j2k0n0α0+2k0[k0n0+ jα0]∆ncos(2β0z+Ω)
By controlling all the perturbed terms, one can define a parameter k[13,15] such that
n j ki jkg
Here ki includes all contributions from the refractive index perturbation whilst kg covers all contributions from the gain perturbation. The parameter k introduced in the above equation is
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known as the coupling coefficient. After a series of simplifications, Eq.(4. 15) becomes )
On substituting the above equation back into the wave equation, one ends up with 0
where the cosine function shown in Eq. (4. 16) has been expressed in phase form. A trial solution of the scalar wave equation could be a linear superposition of two opposing traveling waves such that
jkunz
In order to satisfy the Bragg condition shown earlier in Eq. (4. 11), the actual propagation constant, β , should be sufficiently close to the Bragg propagation constant, β , to make the 0 absolute difference between them much smaller than the Bragg propagation constant. In other words, Such a difference between the two propagation constants is commonly known as the detuning factor or detuning coefficient, δ , which is defined as
δ =β −β0 (4. 21) The trial solution can be expressed in terms of the Bragg propagation constant, i.e.
E(z) =C(z)e−δze−jβ0z +D(z)eδzejβ0z = R(z)e−jβ0z +S(z)ejβ0z (4. 22) where R(z) and S(z) are complex amplitude terms. Since the grating period Λ in a DFB semiconductor laser is usually fixed and so is the Bragg propagation constant, it is more convenient to consider Eq. (4. 22) as the trial solution of the scalar wave equation. By substituting Eq. (4. 22) into Eq .(4. 17), one ends up with the following equation
(R"−2jβ0R'−β02R+β2R+2jβα0R)e−jβ0z +(S"+2jβ0S'−β02S+β2S+2jβα0S)ejβ0z
+2kβ(e2jβ0zejΩ +e−2jβ0ze−jΩ)⋅(Re−jβ0z+Sejβ0z)=0 (4. 23) where R’ and R” are the first- and second-order derivatives of R. Similarly, S’ and S” represent the first- and second-order derivatives of S. With a ‘slow’ amplitude approximation, high-order derivatives like R’’ and S’’ become negligible when compared with their first-order terms. By separating the above equation into two groups, each having similar exponential dependence, one
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can get the following pair of coupled wave equations
− + − j R= jkSe−jΩ Eq. (4. 24) collects all the exp(- jβ0z) phase terms propagating along the positive z direction, whilst Eq. (4. 25) gathers all the exp( jβ ) phase terms propagating along the negative direction. 0z Since δ << , other rapidly changing phase terms such as exp(β ± j3β0z) have been dropped. In deriving the above equations, the following approximation has been assumed
δ Following the above procedures, one ends up with a similar pair of coupled wave equations for a non-zero relative phase difference between the refractive index and the gain perturbation (i.e.
≠0 is the general form known as the forward coupling coefficient and
θ g j i
SR k jk e
k = + (4. 30) is the backward coupling coefficient.
It is contrary to Fabry-Perot lasers, where optical feedback is come from the laser facets.
Optical feedback in DFB lasers is originated from along the active layer where corrugations are fabricated. From the above scalar equation, the couple-wave equation can be established in the general form, which is for one dimensional situation. Following we will discuss two dimensional optical coupling based on above couple- wave theory. For our GaN-based photonic structure, we assume that since the carriers in the InGaN layers are confined in the wall, they do posses a significant in-plane dipole, which can couple to TE mode. Therefore, we centered on TE like mode in square lattice for 2-D case.
2-D couple-wave model
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Preliminary numerical works have been done by Sakai, Miyai, and Noda [16,17]. Here, we cite their papers as references to help us understand the 2-D couple-wave model. The 2-D PC structure investigated here consists of an infinite square lattice with circular air holes in the x and y directions, as shown in Figure 4. 11. The structure is assumed to be uniform in the z direction. We don’t consider the gain effects during calculation. We do calculate the resonant mode frequency as a function of coupling coefficient. The scalar wave equation for the magnetic field Hz in the TE mode can be written as [18]
Figure 4. 11 Schematic diagram of eight propagation waves in square lattice PC structure. where a is the lattice constant,
wc order Bragg diffraction, it will induce 2-D optical coupling and result in surface emission. The coupling constant κ(G) can be expressed as
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where n(G) is the Fourier coefficient of periodic refractive index modulation and λ is the Bragg wavelength given by λ =anav.
In Eq. (4. 32), the periodic variation in the refractive index is included the small perturbation in third term through the Fourier expansion. In the Fourier expansion, the periodic perturbation terms generates an infinite set of diffraction orders. However, as the cavity mode frequency is sufficient close to the Bragg frequency, only the second order diffraction and below can do significant contribution, others can consider to be neglected. Therefore, we focus on diffraction order with m + n ≤2 to discuss. The corresponding coupling coefficient constant κ (j=1, 2, 3) j while considering infinite structure, the magnetic field can be described by the Bloch mode [18],
=
∑ [
+ ⋅]
hGis the amplitude of each plane wave, k is the wave vector in the first Brillouin zone and when it is the Г point, it comes to zero. However, in the case of finite structure, hG is not a constant but a function of vector space. For 2-D case, there are eight propagating waves in PC structure denoted as Rx, Sx, Ry, Sy, F1, F2, F3, F4 showed in Figure 4. 11, those are the amplitudes of four propagating waves in the x, -x, y, -y directions and four propagating waves in Г-M direction, respectively. Those correspond to hG in Eq. (4. 35). Here, we do consider these basic wave vectors along the Г-X directions with κ+ G =β0and Г-M directions withκ+ G = 2β0. The contribution of the higher order waves with κ + G ≥2β0 are considered to be negligible. We should note that the basic waves and higher order waves are partial waves of the Bloch mode, so=