In this chapter, I will briefly introduce the photonic crystal and the photonic band structure. The photonic crystal lasers are also mentioned. Besides, I will state my motivation to the research in the end of the chapter.
1-1 Introduction to the Phonic Crystal Lasers
1-1-1 Introduction to Photonic Crystals
In the world, lots of materials have a specific color. We observe many colors from different goods as the white light is illuminated. There are some mechanisms to decide what color the good may be. One of the causes is the large different absorption coefficient or scattering coefficient of different wavelengths in the visible light region.
Take the sky for example, as the sun light penetrates through the atmospheric layer, Rayleigh scattering occurs through the light's interaction with air molecules and strongly depends on the wavelength (~λ−4); therefore, the shorter blue wavelengths are scattered stronger than longer (red) wavelengths. Hence, we observe the indirect blue light coming from all of the sky. In this example, the high and low densities of the air molecular have a slightly different refractive index; the random distribution of the high and low index regions in the atmosphere results in scattering light in random directions.
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Figure 1-1.1 Picture of the white sun light and the blue sky.
For the colors from the opal [1] and wings of the butterflies [2], scientists find out that they contain a natural periodic microstructure responsible for its natural exhibited color. In other words, the periodic dielectric distribution affects the optical properties of the material. The light scattered in the macrostructure and only certain wavelength can be filtered out. It is the geometry parameter of the microstructure that decides the color. The scale of the geometry must be of the same length-scale as half the wavelength. The structures with periodic index distribution are called photonic crystal. By the applicable fabrication techniques from the semiconductor industry, the photonic crystal structures can be made with a small scale down to visible light region.
Figure 1-1.2 Opal, one of examples of photonic crystal in natural. [1]
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Figure 1-1.3 Phosphor powder of the butterfly with photonic crystal structure.[2]
Photonic crystal structures are one kind of the meta-materials that we can obtain the desired optical properties by fabricating the structure with the well-designed geometry. In general, the structure with periodic dielectric distribution can be called photonic crystal structure. There are three kinds of photonic crystal structures divided by the dimension of the periodic dielectrics distribution; those are one-dimensional (1-D), two-dimensional (2-D) and three-dimensional (3-D) photonic crystals. Figure 1-1.4 shows the illustrations of the 1-D, 2-D and 3-D photonic crystals from left to right. Because of the fabrication difficulty, the three dimensional photonic crystals are less realized and discussed compared to the other ones.
Figure 1-1.4 The illustrations of the 1-D, 2-D and 3-Dphotonic crystals from left to right.
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1-1.2 Photonic Crystal Band Structure
The term of Photonic Crystal has been used after 1987 although it has been studied scientifically for last 100 years. At 1987, Professor Eli Yablonovitch [3] and Saieev John [4] published two papers, which reported the concept of the photonic band gap for the photons in photonic crystal structures. The constructive and destructive interference of light would take place in the photonic crystals. The light with certain wavelength cannot propagate in the structure when the constructive interference regions are not connected. These propagation-forbidden wavelengths form the photonic band gaps. After 1987, the research in photonic crystal increases rapidly. People can control the flow of light by engineering the band structure of the photonic crystals.
In the solid state physics, the motion of the electrons in a crystal will be modulated by the periodic potential function of the nucleuses. A large number of atoms aggregated in a crystal with a specific lattice type will form continuous bands of energy rather than the discrete energy levels of the atoms in isolation. But there are some energy levels that no electrons are allowed to exist with. We call the forbidden regions are the band gaps of the crystal. The behavior of electrons in the crystal can be described with the time-independent Schrodinger equation:
(1.1)
The wavefunction of the electronψ (r) will be influenced by the periodic potential function V(r).
On the other hand, the propagation of the photons in the macrostructure can be affected by the refractive index difference. With the similar concept, the periodic structure with high and low index will form a continuous photonic band of photon
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energy. Photonic band gaps also exist in the band diagram. The behavior of photons in the photonic crystal can be described with the following equation:
(1.2)
The ɛ(r) is the dielectric constant variation function in space. Given the ɛ(r) to describe the photonic structure, we can solve the equation to get the H(r) for a given frequency. For some frequencies, some solutions are not found. It means the light with the frequency cannot exist in the structure, and the forbidden band is formed.
One-dimensional photonic crystal
One-dimensional photonic crystals consist of alternated dielectric layers with different refractive index. Figure 1-1.5 shows the illustration of a1-D photonic crystal structure. The Figure 1-1.6 (a) shows the lattice of a 1-D photonic crystal structure and (b) the corresponding Brillouin zone. The lattice constant is denoted a. The first Brillouin zone and the irreducible Brillouin zone are formed by the intervals [-π /a, + π /a] and [0, +π /a].
Figure 1-1.5 Illustration of a 1-D photonic crystal structure
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Figure 1-1.6 (a) Lattice of a 1-D photonic crystal structure (b) The corresponding Brillouin zone
Figure 1-1.7 is the calculated band structures of 1-D photonic crystal with 13 and 9 alternated dielectric constants and equal thickness. Figure 1-1.8 is the calculated band structures with 13 and 1 dielectric constants. The shadowed regions are the bandgaps of the structures. From the band structures, the band gap size of the 1-D photonic crystal with 13 and 1 dielectric constants are larger than the 13 and 9 one. It implies that the larger index difference of the crystal, the larger band gap size is. In the band structure, the frequency is normalized with the lattice constant. We can obtain the specific modes in the band structure by tuning the lattice constant based on the band diagram. Furthermore, the bandgap size can be controlled by the index difference of the photonic crystal.
Figure 1-1.7 The calculated band structure of 1-D photonic crystal with 13 and 9 dielectric constants. The shadowed region is the bandgap of the structure.
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Figure 1-1.8 The calculated band structure of 1-D photonic crystal with 13 and 1 dielectric constants. The shadowed region is the bandgap of the structure.
The distributed Bragg reflector (DBR) structure is one of the applications of the 1-D photonic crystal. The light with frequencies in the band gap region acts a total reflected region of DBR. Figure shows the simulation result of DBR. The DBR consist of 6-pair SiO2/TiO2. From the simulation result shown in Figure 1-1.10, the reflectivity of the wavelength in 550~700 nm approaches perfect because the light on the region is in the photonic band gap of the 1-D photonic crystal. When the white light incident on the DBR structure, the light with frequencies in the bandgap would be reflected and other portion of light would penetrate though the DBR. It can be applied in the solar cells to enhance the quantum efficiency.
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Figure 1-1.9 Illustration of a DBR structure. The green light would be reflected and other part of white light would pass through the structure.
Figure 1-1.10 Simulation result for the DBR structure. The light with 550 nm~700 nm cannot pass through the structure because it is in the photonic bandgap
region of the 1-D photonic crystal.(Calculated by Kuo, M. Y. RCAS, A.S.)
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Two-dimensional photonic crystal
Figure 1-1.11 and 1-1.12 are the illustrations of 2-D photonic crystals. Figure 1-1.11 is the rod structure with higher dielectric constant. Figure 1-1.12 illustrates a high index membrane with some air holes.
Figure 1-1.11 Illustration of two-dimensional photonic crystals of rod structures
Figure 1-1.12 Illustration of two-dimensional photonic crystals of air holes drilled in a membrane.
Here, we take the triangular lattice photonic crystal as an example. The lattice in real space is shown in Figure 1-1.13 (a). Figure 1-1.13 (b) shows the first Brillouin zone (colored region), which is a smallest enclosed region in reciprocal space formed by the bisectors of the lattice vectors nearby the origin. Figure 1-1.13 (c) shows the irreduced Brillouin zone of triangular lattices (colored region). The symmetry points are denoted “Г, M and K”
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Figure 1-1.13 (a) Illustration of triangular lattice two-dimensional photonic crystals (b) The first brillouin zone (colored region) (c) The irreduced Brillouin zone
of triangular lattices (colored region).
The band structure of the 2-D photonic crystal triangular lattice in a suspend membrane structure is shown in Figure 1-1.14. The 3-D plane-wave extension method was used to calculated the band structure of the photonic crystals with 0.3 radius to lattice constant ratio (r/a) and 430 nm lattice constant. The modes above the light line would be leakage modes. All possible states propagate in the air; therefore, we block the regions above the light line with black color in the band structure. Below the light line, the modes are confined to the slab. The first band gap is at the 0.272 to 0.338 normalized frequency. As the r/a ratio is increased, all of the optical modes will shift to higher frequency region. The blue-shift speeds of different optical modes are not identical. Figure 1-1.15 shows that the normalized frequencies of the first band-edges can be changed by varying the r/a ratio. We find that the first bandgap size is also different with different r/a ratio.
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Figure 1-1.14 The band structure for TE modes of a 2-D triangular lattice photonic crystal with 0.3 r/a
Figure 1-1.15 The normalized frequencies of the first band-edges can be changed by varying the r/a ratio.
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1-1-3 Photonic Crystal Lasers
The photonic crystal structure is widely used in the applications such as to improve the light extraction efficiency of LED [5], photonic crystal waveguide [6]
and so on. One of the important applications of photonic crystals is the photonic crystal lasers. A laser consists of a gain material in a resonate cavity. The light goes through the gain material and the energy of light is then amplified. The process of supplying energy for the amplification is pumping. As for the photonic crystal lasers, the structures are always fabricated with high refractive index gain material. The photonic crystals form a resonate cavity for specific wavelengths. The lasing action would be observed when the threshold condition is reached by the optical or electrical pumping.
In recent years, two-dimensional photonic crystals have become popular due to fabrication improvements and the applications of photonic crystal lasers. In 1994, the idea was proposed by using photonic crystal as resonator. In 1999, the first photonic crystal lasers were demonstrated. Generally speaking, the photonic crystal lasers can be divided into two categories, photonic crystal defect lasers and photonic band-edge lasers. We can create a defect by removing some air holes off or fine tune some air hole positions in the photonic crystals. Light with the forbidden frequency of the photonic band structure will localized and resonate in the defect region. The light will resonate in the plane by the effect of photonic crystal and vertically confined by the large index difference. Once the gain overcomes the loss, the lasing action will be observed. Figure 1-1.16 shows the two mechanisms of the light confinement of a 2-D photonic crystal defect laser. This was first demonstrated by the group of O. Painter [7]. The 2-D photonic band gap and vertical index guiding provide a low loss resonance cavity of the laser. Figure 1-1.17 shows one of examples of a defect laser.
The light resonates in the defect region and achieved lasing. The mode profile of the lasing mode is also shown in the Figure 1-1.17 (b).
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Figure 1-1.16 The configuration of the two-dimensional photonic crystal defect laser[7]
Figure 1-1.17 (a) The SEM image of the defect region. (b)The simulated mode profile distribution of the lasing mode. (c) The light-in light-out curve of the defect laser. [7]
The modes of the deferent kinds of defect lasers and the different modes in the same kinds of the defect laser would all have different optical properties. People can investigate the modes and pick up some desired modes with some specific polarization or lasing wavelength.
On the other hand, the photonic crystal band-edge laser has no physical-defined cavities. They utilize the flat band near the band edge in the photonic band diagram.
The group velocity of light approaches zero there and the lasing action is expected.
There is one band-edge laser shown in Figure1-1.18 [8]. The triangular lattice photonic crystal without any defect can achieve lasing, which is shown in the Figure 1-1.19(a). Compared with the simulation, the lasing modes are at the Г symmetry points, which is denoted as A and B in Figure 1-1.19 (b). The calculated band structure is also shown in the Figure 1-1.19 (b).
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Figure 1-1.18 The configuration of a photonic crystal band-edge laser [8]
Figure 1-1.19 (a) The lasing spectrum and (b) Band structure of the photonic crystal band-edge laser. [8]
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1-2 Motivation and Overview of Thesis
Photonic crystal lasers attract a lot of attention these years. The size is very small compared with other kinds of lasers such as gas lasers. It can be applied in the dense photonic integrated circuit as a signal source [7]. On the other hand, the goal to design and fabricate the thresholdless laser is another important issue of the photonic crystal laser. We can obtain the laser light with less energy input by the small threshold power photonic crystal lasers.
However, for this kind of semiconductor lasers, the lasing wavelength cannot be altered easily because the lasing wavelength strongly depends on the laser geometry.
It limits lots of application due to the single-wavelength property. From another point of view, the change of geometry in the macro-scale area can be detected by the difference in the lasing wavelength. We can measure the wavelength variation to get the information of the local structure deformation instead of using SEM or other more complex methods.
On the other hand, the polymer and organic devices has been studied widely because the application flexibilities. There are demonstrations in the flexible platform for the light source [9-14], modulators [15-16] and sensors [17-18]. Our goal is to integrate the compactness and flexibility. In this way, the application flexibility of the semiconductor devices will increase largely with a compact size.
In this thesis, we demonstrate the multi-wavelength photonic crystal band-edge laser with a flexible PDMS substrate. The optical properties of the lasing modes will changed when the structure is bent. For different lasing modes of the laser, the response to the bent structure will be different. Furthermore, we perform simulations to the phenomenon. Compared with the simulation result, we can deduce the possible geometry variation of the bent laser and estimate the variation percentage for different bending curvature. In this way, we can characterize this flexible photonic crystal laser system.
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With the flexible laser technology, we can fabricate a single device to get a multi-wavelength photonic crystal laser instead of fabricating lots of photonic crystal devices with different lattice constants. Then, the cost of the multi-wavelength device will be reduced. The application in optical communication will be practicable.
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