In this chapter, the plane wave expansion (PWE) method is introduced. We calculate the band structures with 3-D plane-wave expansion method. Then, the model of the geometry deviation on a bent structure is brought up. After that, we estimate the lattice extend percentage by comparing the simulation and measurement results.
4-1 Plane-Wave Expansion (PWE) Method
We use the plane-wave expansion method to calculate the band structures of the photonic crystals. The plane-wave expansion method will be introduced first.
The behavior of the electromagnetic waves can be described by the following
Where B and D are the displacement and magnetic induction field, E and H are electric and magnetic field, and ρ and J are free charge and current density. We
47 Where r is direction vector and t is time.
Then we take the both equation 4.4 and equation 4.5 for curls and the equation 4.5 is divided by ( )r on both sides, then we get equations 4.6 and 4.7: and 4.7 by equation 4.4 and 4.5 respectively, and we get 4.8 and 4.9
2
For mathematics convenient, we introduce the complex forms of E and H
( , ) ( )
tE r t E r e
(4.10)( , ) ( )
tH r t H r e
(4.11)48
From equations 4.14 and 4.15, the operator in this equation is Hermitian operator.
It implies eigenvalues (or frequency) are real and the eigenvectors (or harmonic mode) are orthogonal to each other with different eigenvalues. In addition, another property is scaling properties. Because master equation is scaling invariant, there is no fundamental constant with the dimensions of length for photonic crystal.
There are several ways to calculate photonic band structure. The most common way is plane wave expansion method. Consider equations 4.14 and 4.15:
( ) ( )
2( )
E
E r E r
c
(4.14)
49 expressed as a plane wave times a function with the periodicity of the Bravais lattice:
( ) ( )
i k rApplying equation 4.17, 4.18 and 4.19 to equation 4.14 and 4.15, we can get the following equation:
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'
' ' ' ' 2 '
( )( ) {( )
k( )}
k k( )
G
G G K G K G H G H G
(4.21) Solve these two equations numerically, and plot the dispersion relation of
and k to get the photonic band structure.
In this thesis, all the band structure calculations are performed with PWE method by the commercial software R-soft.
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4-2 Optical Modes of the Band-edge Lasers
In order to investigate the optical modes of the triangular lattice photonic crystal lasers on a flexible PDMS substrate, we calculate the corresponding band structure of the photonic crystal by 3-D plane-wave expansion (PWE) method for TE-like modes.
Figure 4-2.1 is the calculated band structure with 0.25 radius to lattice constant ratio.
Compared with the suspend membrane structure, the light core shifts to the lower frequency region because the PDMS substrate has higher refractive index than air.
The band-edge lasing modes are likely to occur around the high-symmetry points of the band structure. The flat dispersion curve near the band-edge implies a low group velocity of light and strong localization. Hence, the light-matter interaction can be enhanced and lasing action is expected. Figure 4-2.2 shows the group velocity of light with different in-plane wave vector k of Г to M point. At the symmetry point M, the group velocity approaches zero and the lasing action is expected.
At chapter 3, we have measured two lasing modes with the normalized frequencies around 0.249 and 0.264. Comparing the measurement with simulation, we identify the M1 and the K1 in the band structure are the operation modes of the photonic crystal lasers.
Figure 4-2.1 The TE-like mode band structure of the triangular lattice photonic crystals with 3-D plane-wave expansion method. The K1 and M1 are identified as the
lasing modes.
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Figure 4-2.2 The group velocity versus different wave vector k. The group velocity approaches zero for modes near the M1.
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4-3 Model for Geometry Variation in the Flexible Photonic Crystal Laser System
In the experiment, we observed that the lasing wavelength of the band-edge laser red-shifts as the bending curvature is increased. Here, we attribute the red-shift to the lattice distortion and the change of the PDMS refractive index.
The lasing wavelength of the photonic crystal lasers strongly depends on the laser geometry. We can modify the lattice constant to make the device achieve lasing at specific wavelength. In this flexible laser system, as we bend the structure along Г-M direction, the red-shifts in lasing wavelength are observed. These red-shifts imply that the geometry of the laser would be changed when the structure is bent. In other words, we can measure the change in lasing wavelength to presume the possible mechanism of the geometry variation. By comparing the measurement with the simulation, we can quantize the geometry variation in the flexible laser system.
We create a model for the geometry variation in the flexible laser system. Figure 4-3.1 and Figure 4-3.2 illustrate the photonic crystals on a plate and on a bent PDMS substrate relatively. We assume that the lattice constant in the Г-K direction will increase as the structure is bent; the lattice constant in the Г-M direction would remain the same. Figure 4-3.3 shows the triangular lattice photonic crystals and the Г-K and Г-M directions.
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Figure 4-3.1 The illustration of the triangular lattice photonic crystals with a plate PDMS substrate.
Figure 4-3.2 The illustration of the triangular lattice photonic crystals with a bent PDMS substrate.
Figure 4-3.3 The illustration of the triangular lattice and the Г-K and Г-M directions of the lattice.
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On the other hand, the refractive index of the PDMS substrate will become smaller when the structure is bent. We assume that the volume of the PDMS substrate will increase as the curvature is increased. With the equation 4.22[19], we can obtain the PDMS refractive index at different curvatures by estimate the enlarge volume of the PDMS substrate.
(4.22)
The symbol V in equation 4.22 is denoted as the volume of the PDMS substrate.
Because the negative sign in the equation, the PDMS index will become smaller when the PDMS volume is increased. Figure 4-3.4 shows the refractive index of the PDMS substrate versus the bending curvature.
Figure 4-3.4 The refractive index of the PDMS substrate versus the bending curvature.
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4-4 Simulation for the Variation of the Optical Modes
In the model for the flexible laser system, we assume that the lattice distortion and the change in PDMS index will influence the frequencies of the modes. The lattice constant in the Г-K direction would be increased and the lattice constant in the Г-M direction would remain the same after we bend the structure along the Г-M direction. Figure 4-4.1 illustrates our model for the lattice variation. The dash circles colored in brown denote the extended lattices and the blue circles are the original lattices. In this way, the unit cell of the triangular lattices will change and the Brillouin zone changes as well. Figure 4-4.2 illustrates the deviation of the unit cell of the lattice. The height of the unit cell would not change and the width extends about 20%. Figure 4-4.3 illustrates the deviations of the Brillouin zones with different lattice extension percentage.
Figure 4-4.1Illustration of our model for the lattice variation.
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Figure 4-4.2 Illustration of the unit cell deviation due to the lattice extension
Figure 4-4.3 Illustration of the Brillouin zones with different lattice extension percentage.
When the lattice constant of the photonic crystal is increased in the Г-K direction, the six-fold symmetry of the triangular lattices will be broken and turns out to be four-fold symmetry system. Г, M and K are the conventional symbols to denote the symmetry points of the triangular lattices. However, in this system, we have to use five symbols to point out the symmetry points in the four-fold symmetry of the extended triangular lattices. Here we use “Г, M ,M’, K and K’ and Figure 4-4.4 shows the four fold symmetry Brillouin zone and the new irreduced Brillouin zone. The new irreducible zone is light blue shaded. Then, we calculate the band structure with the k path along the new Brillouin zone. In this way, the behavior of the waves in the lattice-extended lattice can fully characterized by their behavior in the new irreduced Brillouin zone. Figure 4-4.5 is the calculated band structure with 3-D PWE method.
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Figure 4-4.4 Illustration of the four fold symmetry Brillouin zone and the irreduced Brillouin zone with symmetry points denoted Г,M ,M’, K and K’ .
Figure 4-4.5 The TE-like mode band structure of the lattice-extended triangular lattice photonic crystals with the K path along the new irreduced Brillouin zone.
We perform 3-D PWE simulations to the deviation of the band-edge modes with different lattice extension percentage, which is from 0% to 6% with fixed air hole shape and radius. For example, for photonic crystals with 400 nm lattice constant, the 6% lattice extension percentage means the lattice constant in the Г-K direction is increased to 424 nm.
The simulation result of the photonic crystal band-edge laser with 0.28 r/a ratio is shown in Figure 4-4.6. We find that the band-edge modes on the first band red shifts
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with different speeds. When the lattice in the Г-K direction is increased, the detergency of the fist M mode (M1) will break and significantly split into two modes M1 and M’1.When the lattice extends 6%, the normalized frequency of M1 would change from 0.2468 to 0.2456 (M) and to 0.2368 (M’). The K1 mode does not split in the range of the calculations. The normalized frequency of the K1 changes from 0.271 to 0.264.
Figure 4-4.6 Simulation results to the red-shift of the band-edge mode. The M1
mode split into two modes with different red-shift speeds.
The lasing oscillation of the M1 band-edge mode is in three M directions of crystals. Figure 4-4.7 shows the illustration of the three lasing oscillations of the M1
band-edge mode [20]. Figure 4-4.8 is the Hz field mode profiles of the M1 band-edge mode with the lasing oscillations along three M directions.
Figure 4-4.7 Illustration of the lasing oscillations of the M1 band-edge mode in real space. Each oscillation is formed by the coupling between two
counter-propagating waves.
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Figure 4-4.8 The Hz mode profile of the M1band-edge mode with the lasing
oscillations along three M directions.
For the non-extended photonic crystals, these three lasing oscillations have the same normalized frequency. However, when the lattice is extended in Г-K direction, the six-fold symmetry of the triangular lattices is broken. The lasing oscillations in the three different directions are suffered different degrees of the influence by the lattice extension. Hence, the degeneracy of the M1 band-edge mode is broken and split into two modes, M1 and M’1. As a sequence, the lasing oscillation of the M1 band-edge mode is along the Г-M directions. On the other hand, the M’1 mode is composed of the other two lasing oscillation directions. The oscillation period in the two directions becomes longer after the lattice is extended.
On the other hand, the K1 band-edge mode does not split significantly as the M1 band-edge mode. We can find some reasons from the feedback mechanisms at K1 mode[20]. Figure 4-4.9 is the illustration of the two lasing oscillations of the K1 band-edge mode. Each oscillation of the K1 band-edge mode is formed by a sum combination of three nonparallel wave vectors which form closed loops [20]. The mode profiles of the K1 mode are also shown in Figure 4-4.10. It indicates that the influence of the lattice extension to these modes is the same; therefore, the K1 mode does not split when the lattice in Г-K directions is extended.
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Figure 4-4.9 Illustration of the lasing oscillations of the K1 band-edge mode in real space. Each oscillation is formed by a sum combination of three nonparallel wave
vectors which form closed loops.
Figure 4-4.10 The Hz field K1 mode profiles. The K1 mode is composed of these two profiles.
The reduction of the PDMS index will cause the lasing wavelength blue-shift.
Figure 4-4.11 shows the M1 and K1 band-edge modes would blue-shifts when the PDMS refractive index becomes lower.
Figure 4-4.11 The band-edge modes blue-shift when the PDMS refractive index becomes lower.
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From Figure 4-3.4, we can obtain the relationship between the refractive index of the PDMS substrate and the curvature of the structure. Hence, we can further find out the variation of the M1 and K1 band-edge modes by the PDMS index change with different bending curvatures. Figure 4-4.12 shows the M1 and K1 modes blue-shifts with the increase of the bending curvature.
Figure 4-4.12 The blue-shifts of the band-edge modes caused by the PDMS index change at different bending curvatures.
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4-5 Experiments for the Simulation Accuracy
To verify the accuracy of the simulation, we make the devices with different lattice extension percentage on a PDMS substrate. The designed lattice extension percentage is 0%, 0.2%,0.4%,0.8% and 1%. Figure 4-5.1 shows an array of photonic crystal band-edge lasers with different lattice extension percentages in the same row and different lattice constant in the same column. Figure 4-5-2 is the magnified SEM image of the photonic crystal structure.
Figure 4-5-1 A array of photonic crystal band-edge lasers to verify the simulation
Figure 4-5-2 The magnified SEM image of the photonic crystal structure. (angle view.)
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Figure 4-5-3 shows the comparison between the measurement and simulation results. The simulation is performed with 0.25 r/a ratio and 430 nm lattice constant.
The simulation result is shown in Figure 4-5-3 with a blue line. The observed lasing mode is at the K1 symmetry point with 0.2633 normalized frequency. As the lattice extension percentage is increased, the lasing wavelength red-shift. The tendency of the red-shift in lasing wavelength agrees the simulation well. It implies that the simulation is correct.
Figure 4-5-3 The comparison between the measurement and simulation results.
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4-6 Simulation Fitting for the Experimental Results
We observe the lasing wavelength red-shifts as the bending curvature is increased. Based on our model for the lattice variation, the observed red-shift in frequency is resulted from the combination of the lattice distortion and the PDMS refractive index change. In this section, we fit the red-shifts data with the simulation result. Figure 4-6.1 shows the fitting results of the K1 mode. The red-line shows the simulated frequency due to the lattice distortion. The frequency of the K1 mode decreases as the curvature is increased. The blue-line shows the simulated frequency of the K1 mode which increases as the PDMS index is decreased. The green line is the linear fitting results of the measured data. The red-shift caused by the lattice distortion is compensated by the blue-shift of the PDMS refractive index. In other words, we do not observe the larger red-shift because the mode is also influenced by the index effect. From the fitting result, as the bending curvature is increased to 0.06 mm-1, the lattice would extend about 0.2% in the Г-K direction.
Figure 4-6.1 The fitting results of the K1 mode. The lattice extend about 0.2% in the Г-K direction when the curvature is increased to 0.06 mm-1
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Figure 4-6.2 shows the fitting results of the M’1 mode. In the simulation, the M1
mode would split into two modes; these modes have different red-shift speeds. In the measurement, we also observed two red shift speed of the M1 mode. We also attribute the red-shift to the lattice distortion and the change of the PDMS index. In Figure 4-6.2, the red-line shows the simulated frequency due to the lattice distortion. The frequency of the M’1 mode decreases as the curvature is increased. The blue-line shows the simulated frequency of the M’1 mode which increases as the PDMS index is decreased. The green line is the linear fitting results of the measured data. The red-shift speed of the measured data is not as high as the simulation result to the lattice distortion because the index effects compensate the red-shift. Comparing the simulation result and the measurement, we conclude that when the curvature of the laser is increased to about 0.06 mm-1, the lattice would extend about 0.2% in the Г-K direction. This estimation is almost the same with the fitting result of the K1 mode.
Figure 4-6.2 The fitting results of the M’1 mode. The lattice extends about 0.2%
in the Г-K direction when the curvature is increased to 0.06 mm-1
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The agreement of the estimations indicates that the lattice would extend in the Г-K direction when the structure is bent along Г-M direction. Hence, we quantify the structure deviation by the simulation. The lattices extend in the Г-K direction about 0.2 percent when the curvature of the device is 0.06 mm-1. This small change of the structure cannot be easily detected by the SEM or other measurement tools; however, the lasing wavelength can provide information about this geometry variation. We can apply this kind of device as a curvature sensor.
4-7 Conclusion
In this chapter, we simulated the photonic crystal band structures with 3-D-PWE method. We identified the two lasing modes with 0.249 and 0.265 normalized frequency, which are the band-edge modes corresponded to M1 and K1 in the band structure. Then, the model of the geometry deviation on a bent structure is brought up.
We assume that the lattice will have a perturbation due to the structure deformation.
The change of the PDMS refractive index is also considered in the simulation.
Comparing with the simulation, we estimate that the lattice constant in Г-K direction extend about 0.2 percent when the curvature is increased from 0 to 0.06 mm-1.
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Chapter 5 Summary and Future Works
In this thesis, the triangular lattice photonic crystal band-edge lasers are fabricated on a flexible polydimethylsiloxane (PDMS) substrate. The lasing wavelength red-shift when the structure is bent. We quantify the geometry deviation by fitting the red-shift range with the PWE simulation results.
The fabrication procedures of the flexible photonic crystal lasers are introduced.
The index of the PDMS substrate is low enough for the optical light confinement in vertical direction. The structure achieve lasing and two lasing mode are identified.
When the structure is bent along Г-M direction, the lasing wavelength red-shifts. The red-shift speeds of different lasing modes can provide us the information about the structure deviation. By 3-D PWE method, we can quantify the geometry deviation by fitting the red-shift data without using SEM. The lattice of the photonic crystal band-edge laser would extend 0.2% in the Г-K direction when the bending curvature is about 0.06 mm-1.
The flexible laser can be a local curvature sensor by observing the change in lasing wavelength. It also can be a novel multi-wavelength light source used in integrated photonic circuit with a very compact size.
In the future, we will continue to study this project in several directions:
1. We will demonstrate the sensitivity tunable index sensor. The mode distribution of the photonic crystal lasers would be different when the curvature is changed. This would alter the sensitivity to outer index changes.
We can find the best curvature of the device by finding the maximum value of the quality value multiply the sensitivity.
2. We will demonstrate the bio-applications to detect the behaviors of bio molecular in vitro.
3. We will demonstrate the micrometer scale photonic integrated circuit in the flexible platform.
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