Chapter 1 Introduction
II. 2D PC Band-edge lasers
1.4 Outline of the thesis
This essay has been organized in the following way. The first section of the paper will examine the history of nitride-based materials. Chapter 2 begins by laying out the theoretical dimensions of the research, the couple-wave theory and oscillation feedback mechanism. Chapter 3 describes the wafer preparation and fabrication of PCSELs. Chapter 4 describes the design, synthesis, characteristics and simulation of PCSELs. The last chapter assesses the conclusion and future work.
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References
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2. S. Nakamura, T. Mukai, and M. Senoh, Appl. Phys. Lett.,64, 1687 (1994)
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Sugimoto, and H.Kiyoku, Appl. Phys. Lett., 70, 868 (1997) 4. S. Nakamura, Science, 281, 956 (1998)
5. E.M. Purcell Phys. Rev. 69, 681 (1946)
6. C. M. Lai, H. M. Wu, P. C. Huang, S. L. Peng, Appl. Phys. Lett., 90, 141106, (2007)
7. P. R. Berman, New York:Academic, (1994)
8. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O`Brien, P. D. Dapkus, I. Kim, Science, 284, 1819, (1999)
9. H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, Y.
H. Lee, Science, 305, 1444, (2005)
10. Y. S. Choi, K. Hennessy, R. Sharma, E. Haberer, Y. Gao, S. P. DenBaars, C. Meier, Appl. Phys. Lett., 87, 243101, (2005)
11. M. Imada, S. Node, A. Chutinan. and T. Tokuda, Appl. Phys. Lett., 75, 316, (1999) 12. D. Ohnishi, T. Okano, M. Imada, and S. Node, Opt. Exp., 12, 1562, (2004)
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Chapter 2
Fundamentals of Photonic Crystal Surface Emitting Lasers
Introduction
A considerable amount of literature has been published on photonic crystal surface emitting lasers utilizing a 2D distributed feedback (DFB) mechanism [1-4]. The device features single longitudinal and transverse mode, lasing with large area and narrow beam divergence. To calculate photonic band-gap and the distribution of electric or magnetic field, there have been many theoretical analysis and methods developed, such as 2D plane wave expansion method [2,5] (PWEM), finite difference time domain
[6,7] (FDTD), Transfer Matrix method and Multiple scattering method, etc. To optimize
PCSELs structure, most experimental data agree with PWEM and FDTD. But there are some limitations while using these theoretical methods. 2D PWEM only applies to the infinite structure that is contradictory to actual device. FDTD method consumes numerous computer memories to simulate the real structure. Thus, we use the simple and convenient method couple-wave theory to describe the oscillation mechanism in PCSELs structure. Preliminary work on couple-wave theory was undertaken by Kogelnik and Shank [8], they presented the couple-wave analysis of distributed feedback lasers near the Bragg diffraction. As a result, we will focus on the fundamental of couple-wave theory.
2.1 Couple-wave theory
[8,9]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
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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.
To get started, we recall the fundamental wave equation to help us understand it. In a homogeneous, source-free and lossless medium, any time dependent harmonic electric field satisfy the vector wave equation
2 0
2 + 02 =
∇ Er k n Er
(2.1) 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 (2.2)
Consider a multi-dielectric stack in which periodic corrugations are formed along one boundary as illustrated in Fig 2.1.
Figure 2.1 General multi-dielectric layers show the perturbation of refractive index and amplitude gain. Z1(x) and Z2(x) are two corrugated functions.
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The material complex permittivity in each layer is denoted asεj while g and Λare 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' (2.3) where w is the angular frequency andε’ is the complex permittivity. When the radiation frequency is sufficiently close to the resonance frequency, eqn (2.3) becomes
⎟⎟⎠ where n(z) andα(z)are the refractive index and the amplitude gain coefficient, respectively. Within the grating regiondx≤x≤dx+g, perturbation is considered so the refractive index and gain coefficient can be expressed in a Fourier form as
n(z)=n0+Δncos(2β0z+Ω) (2.5-1)
and
α(z)=α0+Δαcos(2β0z+Ω+θ) (2.5-2) Here, n0 andα0are the steady-state values of the refractive index and amplitude gain, respectively. nΔ and Δ are the amplitude perturbation terms, α β is the 0
propagation constant and Ω is the non-zero residue phase at the z-axis origin. In the eqn (2.5-2), 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, lossless waveguide at an angle of Φ as shown in Fig. 2.2. The propagation constant of the wave is assumed to beβ . 0
Figure 2.2 A simple model used to explain Bragg conditions in a periodic waveguide.
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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 the reflected wavelets must be a multiple of 2π. In other words,
β0(AB+BC)=β0(2ΛsinΦ)=2mπ (2.6) where m is an integer. If the incident wave is now approaching more or less at a right angle to the wavefront (i.e. Φ≈π2), eqn (2.6) becomes
2β0Λ =2mπ (2.7) This is known as the Bragg condition andβ becomes the Bragg propagation 0 constant. The 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 [8]. Rearranging eqn (2.7) gives
= Λ where λB and w are the Bragg wavelength and the Bragg frequency, respectively. B From eqn (2.8), 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<<
Δ , Δα <<α0 (2.9) Substituting eqn (2.5) into (2.4) using the above assumption, generates
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By controlling all the perturbed terms, one can define a parameter k[8,10] such that
n j ki jkg
Here k includes all contributions from the refractive index perturbation i whilstk covers all contributions from the gain perturbation. The parameter k g introduced in the above equation is known as the coupling coefficient. After a series of simplifications, eqn (2.12) becomes
)
On substituting the above equation back into the wave equation, one ends up with 0
where the cosine function shown in eqn (2.14) has been expressed in phasor 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 eqn (2.8), the actual propagation constant,β , should be sufficiently close to the Bragg propagation
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constant, β , to make the absolute difference between them much smaller than the 0 Bragg propagation constant. In other words,
0
0 β
β
β − << (2.18) 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 (2.19) The trial solution can be expressed in terms of the Bragg propagation constant, i.e.
z 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 eqn (2.20) as the trial solution of the scalar wave equation. By substituting eqn (2.20) into eqn (2.15), one ends up with the following equation 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 can get the following pair of coupled wave equations
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Equation (2.22) collects all the exp(- jβ ) phase terms propagating along the 0z positive z direction, whilst eqn (2.23) gathers all the exp( jβ ) phase terms 0z propagating along the negative direction. Since δ << , other rapidly changing β phase terms such as exp(± 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. θ ≠0) such that is the general form known as the forward coupling coefficient and
θ g j i
SR
k jk e
k = +
(2.28) 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
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square lattice for 2D case.
2.2 2D couple-wave model
Preliminary numerical works have been done by Sakai, Miyai, and Noda [12,13]. Here, we cite their papers as references to help us understand the 2D couple-wave model.
The 2D PC structure investigated here consists of an infinite square lattice with circular air holes in the x and y directions, as shown in Fig 2.3. 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 [14] coefficient of the periodic refractive-index modulation and λ is the Bragg wavelength given by λ =anav. In the eqn (2.31), we set α,αG<<β0,nG<<n0. We do consider Γ point, in which when it is satisfy the second order Bragg diffraction , it will induce 2D optical coupling and result in surface emission. The coupling constant )κ(G can be expressed as
where )n(G is the Fourier coefficient of periodic refractive index modulation and λ is the Bragg wavelength given by λ =anav.
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Figure 2.3 Schematic diagram of eight propagation waves in square lattice PC structure
In eqn (2.31), 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 jκ (j=1, 2, 3) are denoted as
κ1=κ(G)G =β0
0
2 κ( ) 2β
κ = G G = (2.33)
0
3 κ( ) 2β
κ = G G =
while considering infinite structure, the magnetic field can be described by the Bloch mode [14],
=
∑ [
+ ⋅]
G
G j k G r
h r
Hz( ) exp ( ) (2.34)
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 2D case, there are eight
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propagating waves in PC structure denoted as Rx, Sx, Ry, Sy, F1, F2, F3, F4 showed in Fig 2.3, 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 eqn (2.34). 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 they have the same eigenvalue β for specific resonant cavity mode.
The magnetic field in this case can be rewritten as
Hz =Rx(x,y)e−jβ0x +Sx(x,y)ejβ0x +Ry(x,y)e−jβ0y +Sy(x,y)ejβ0y
Put eqn (2.35) and eqn (2.31) into the wave eqn (2.30), and comparing the equal exponential terms, we obtain eight wave equations
(β−β0)Rx+κ3Sx−κ1(F2+F4)=0 (2.36a)
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These derivations illustrate the coupling among the propagating waves in square lattice PC structure. We take eqn (2.36) for example. It describes the net wave superposition along the x-axis, including the coupling of two waves propagating at opposite directions, Rx and Sx, with a coupling coefficient κ . And the coupling of 3
higher order waves F2 and F4, with a coupling coefficient κ1. The coupling κ 3
provides the main distributed feedback. From above derivations, we know that the orthogonal couplings occur via intermediate coupling of the basic and higher order waves. One thing we should noted that the coupling coefficient κ2 does not exist in eqn (2.36). In the numerical view which describes the basic waves directly couple to orthogonal directions. This physically can be explained in this way: ways of TE modes have their electric field parallel to the PC plane, so that the electric field directions of the two waves propagating in the perpendicular directions are orthogonal to one another, hence the overlap integral of the two waves vanishes.
By using 2D plane wave expansion method, the band dispersion curves for TE like modes in Fig 2.4 can be obtained. The condition is limited at the vicinity near Bragg frequency.
Figure 2.4 Dispersion relationship for TE like modes, calculated using the 2D PWEM
By solving eqn (2.36) . The cavity mode frequency can be obtained. There are three
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The frequencies wA and wB at the lower band-edges are non-degenerated, while two of the frequencies wc and wD at higher band-edge are degenerated. In the physical meaning, the resonant mode symmetries are different. The resonant mode at band-edge C and D are symmetric, which allows this mode to couple to external field more easily. The characteristics of these resonant modes are essentially leaky [16]. The resonant mode at band-edge A and B are anti-symmetric, resulting in less coupling to the external field. Therefore, the quality factor for band-edge A and B are higher than C and D. It is expected that the lasing behavior is occurred at either A or B. However, band B is flat around Г point in the Г-X direction. Light at band-edge B can couple to leaky mode with a wave vector slightly shifted from Г point, resulting in coupling to the external field. Thus, band-edge B becomes leaky.
So far we have understood the fundamental couple-wave theory and how to calculate their cavity mode frequency. For our GaN-based PCSELs, we design the triangular lattice with TE-like mode to tell the differences between square lattice structure.
Therefore, in next section, we develop a new model to explain the DFB feedback mechanism based on couple-wave theory. According to our measurement, we found that the lasing action occurred at Г1, K2, M3 point band-edges. We tried to solve the wave equations at these band-edges and derive the coupling coefficients. All of detail will be described numerically in next section.
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2.3 Couple-wave model for triangular lattice PCSELs
Light at the photonic band-edge has zero group velocity and forms a standing wave due to 2D DFB effect. Laser oscillation is expected to occur at any band-edge, if the gain threshold is achieved. Therefore, we focus on the coupling waves at Г1, K2, M3 band-edges according to our lasing behaviors.
I. Г1 numerical results
The 2D PC structure investigated here consists of an infinite triangular lattice with circular air holes in the x and y directions, as shown in Fig 2.5. The structure is assumed to be uniform in the z direction. We don’t consider the gain effects during calculation. While considering infinite structure, the magnetic field can be described by the Bloch mode [14],
Figure 2.5 Schematic diagram of six propagation waves in triangular lattice for Г1 point
put eqn (2.38) and eqn (2.31) into the wave eqn (2.30), and comparing the equal
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exponential terms, we obtain six wave equations
3 4 of individual light waves propagating in the six equivalent Г-M directions: 0°, +60°, +120°,+180°, +240°, and +300° with respect to the x-axis. κ1, κ2, and κ3 are the coupling coefficients between light waves propagating at 60° to each other (H1 and H2, H2 and H3,and so on), at 120° (H1 and H3, H2 and H4, and so on), and at 180° (H1 and H4, H2 and H5, and so on), respectively. δ is the deviation of the wave number β (expressed as 2πν/c, where ν is the frequency and c is the velocity of light) from the fundamental propagation constant β0 (equal to 4π 3a, where a is the lattice
constant) for each cavity mode, and expressed as δ =(β2 −β02)/2β0, α is the corresponding threshold gain.
By solving eqn (2.39a-e), a cavity frequency ν for each band-edge mode and the corresponding threshold gain α for a given set of coupling coefficients, κ1, κ2, and κ3, can be obtained. When only the cavity mode frequencies are required, the derivation terms and the threshold gain α in eqn (2.39a-e) can be set to zero, and the individual cavity frequencies can then be derived as follows:
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where, c is the velocity of a photon in vacuum, and neff is the effective refractive index of the device structure. There are four cavity mode frequencies, ν1 −ν4 , which correspond to the four band-edge, including two degenerate modes ν and 2 ν . 4 Once the cavity mode frequency at the individual band-edges can be obtained, we can derive the coupling coefficientsκ1, κ2, and κ3 from eqn (2.40a-d) as follows: device parameters, we can determine which kind of DFB mechanism provide the major significant contribution to support the lasing oscillation.
II. K2 numerical results
The 2D PC structure investigated here consists of an infinite triangular lattice with circular air holes in the x and y directions, as shown in Fig 2.6. The structure is assumed to be uniform in the z direction. We don’t consider the gain effects during calculation. While considering infinite structure, the magnetic field can be described by the Bloch mode [14],
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Figure 2.6 Schematic diagram of three propagation waves in triangular lattice for K2 point
put eqn (2.42) and eqn (2.31) into the wave eqn (2.30), and comparing the equal exponential terms, we obtain three wave equations :
)
where, H1, H2, H3 express the envelope magnetic field distributions of individual light waves propagating in the three equivalent Г-K directions: 0°, 120°, 240° with respect to the x axis. κis the coupling coefficient between light waves propagating at 120° to each other (H1 and H2, H2 and H3, H1 and H3), δ is the deviation of the wave number β (expressed as 2πν/c, where ν is the frequency and c is the velocity of light) from the fundamental propagation constant β0 (equal to 8π/3a, where a is the lattice constant) for each cavity mode, and expressed as
0 2 02)/2
(β β β
δ = − , α is the corresponding threshold gain.
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By solving eqn (2.43a-c), a cavity frequency ν for each band-edge mode and the corresponding threshold gain α for a given set of coupling coefficients, κ can be obtained. When only the cavity mode frequencies are required, the derivation terms and the threshold gain α in eqn (2.39a-e) can be set to zero, and the individual cavity frequencies can then be derived as follows:
)
where, c is the velocity of a photon in vacuum, and neff is the effective refractive index of the device structure. There are two cavity mode frequencies, ν1,ν2 , which correspond to the two band-edge, including one degenerate modes ν . Once the 2 cavity mode frequency at the individual band-edges can be obtained, we can derive the coupling coefficientsκfrom eqn (2.44a,b) as follows:
From the 2D couple-wave model, we know light at K2 band-edge can couple to each other and form the triangular feedback close loop via DFB effect.
III. M3 numerical results
III. M3 numerical results