Structure
To characterise the behaviour and simulate the reflection spectra of these binary biharmonic structures we used rigorous coupled wave analysis (RCWA). Firstly, we need to choose the period of the coupler grating, which will determine the centre frequency of the band gap. We want the centre of the band gap region in the near-infrared wavelength range, around 1.5µm. Consider normal incidence, θ= 0, Eq. 4.12 becomes
kSP P =±dKcplr (4.14)
wherekx has been replaced with kSP P. Equivalently, form Eq. 4.9 ω c r 12 1+2 =±d 2π Λcplr (4.15) Λcplr =λ0 r 1+2 12 (4.16) where ω/c has been replaced with 2π/λ0 and λ0 is the free space wavelength.
With the grating in air,1 = 1, for gold the dispersion is wavelength dependent,
at 1.5 µm, 2 = −100.05 +i3.5545 [102]. Taking the real part of Eq. 4.16 one
gets Λcplr = 1.492µm. To make things simple, we will use a period of 1.5 µm (SPP wavelength of 1.508 µm) which corresponds to a Bragg grating period of 0.750µm.
To begin, we first look at the reflection spectra for the Bragg and coupler gratings individually for four different groove depths; 50 nm, 100 nm, 150 nm and 200 nm. Each has a 50% fill-factor and the wavelength range is 0.5 µm to 2.5 µm. The reflection spectra are calculated for a complete range of incident angles; from normal to 90◦ for TM polarisation. The results are presented in Fig. 4.15 where the x-axis is the parallel wave vector component of the incident light,
Period 750 nm Groove Depth 50 nm Period 1500 nm
Groove Depth 100 nm
Groove Depth 150 nm
Groove Depth 200 nm
Figure 4.15: Calculated refection spectra for the Bragg and coupler grating structures with respective period values of 750 nm and 1500 nm. Both structures have a fill-factor of 50% with the incident radiation TM polarised. The grating groove depth is varied; 50 nm, 100 nm, 150 nm and 200 nm. The dispersion of a SPP along a gold-air interface is plotted over the reflection spectra at each of the grating wave vector scattering points as a white dashed line.
kx =k0sin(θ), the left hand y-axis is the angular frequency while the right hand
y-axis is the wavelength of the light. These graphs represent the complete light cones as shown previously in Fig.s 4.10, 4.12 and 4.13. In each of the plots the dispersion relation of a SPP propagating along a gold-air interface is overlaid as a white dashed line.
Good agreement is achieved between the shape of the reflection bands and the overlaid SPP dispersion relation and the behaviour is as expected from Fig. 4.12, i.e. the reflection bands should follow the SPP dispersion and curved away as they approach a crossing point. The blue region (very low reflectivity) at the top of each plot corresponds to the wavelength region where the gold is naturally very lossy and so the light is strongly absorbed there. For both grating structures the position of the bands is clear, but, the coupling is very weak i.e. the reflectivity is relatively high which corresponds to low absorption. The band gaps at the crossing points are small and not distinct. Although the bands do appear to get “flatter” as the groove depth is increased.
Of course, the band gap of interest is outside the light cone of the 750 nm period grating (not visible with the single period grating structure). The other area of interest is the second order band gap of the coupler grating (1500 nm) i.e. the first crossing point in the centre region of the dispersion plot. This is where the band gap of the biharmonic structure will be. The band gap produced by the the single period structure is very small and the coupling is very weak.
Next we examine the reflection spectra for the biharmonic structure where the Bragg and coupler grating have been combined. For this, we look at a slightly narrower wavelength range i.e. from 1µm to 2.5µm, in order to zoom in on the band gap region. The same full angle range is covered to 90◦ along with the same four groove depths; 50 nm, 100 nm, 150 nm and 200 nm and of course, all for TM polarisation. Four different amplitude ratios of the coupler to Bragg grating are covered;A2/A1 = 0.51, 0.75, 1.0 and 1.5. The simulation results are
presented in Fig. 4.16 with the groove depth increasing down the columns and the A2/A1 ratio increasing across the rows. Along with the sixteen reflection
dispersion plots, the four reflection plots of the coupler grating (A2/A1 = 0) are
replotted from Fig. 4.15 in the first column of the figure for easy reference and comparison.
Firstly, for an amplitude ratio of 0.51, the behaviour is not too unlike that of the coupler grating in the first column. The overall shape of the bands is similar, but, the biharmonic structure does exhibit stronger coupling in places. This similar behaviour is expected as the biharmonic grating profile is very close to that of the single period coupler structure with only a very narrow groove in the grating being the difference (Fig. 4.14a). For all the other dispersion plots, a band gap is easily resolved although only the, low frequency, arm of the gap is clearly obvious.
The higher frequency arm of the gap is very weakly coupled and its presence is not easily resolved in the reflection plots due to the large colour range. By examining the reflection spectra directly, the coupling of the higher frequency modes can be seen.
The general trend in Fig. 4.16 is that increasing the amplitude ratio A2/A1
increases the size of the band gap, as expected from [95]. The band gap width is also proportional to the groove depth; for shallow gratings (50 nm) no band gap
A 2 /A1=0.5 1 A 2 /A1=0.7 5 A 2 /A1=1.0 A 2 /A1=1.5 0 A 2 /A1=0 Groo ve Dep th 200 nm Groo ve Dep th 150 nm Groo ve Dep th 100 nm Groo ve Dep th 50 nm
Figure 4.16: Reflection dispersion plots for the binary biharmonic grating structure for a range for of groove depths (50 nm, 100 nm, 150 nm and 200 nm) and a for range
ofA2/A1 amplitude ratios (0.51, 0.75, 1.0 and 1.5). The first column of images are the
same as the second column of Fig. 4.15 i.e. the dispersion plots for the coupler grating plotted here with a slightly narrower wavelength range.
0.6 0.8 1 1.2 1.4 0 50 100 150 200 250 300 350 400 450 A2/A1
Band Gap Width
Δλ (nm) 50 nm 100 nm 150 nm 200 nm 50 100 150 200 0 50 100 150 200 250 300 350 400 450 Groove Depth (nm)
Band Gap Width
Δλ (nm) A2/A1=0.75 A2/A1=1.00 A2/A1=1.50 (a) (b)
Figure 4.17: Summary of how the band gap width of the biharmonic structure, at
normal incidence, varies with increase in amplitude ratio (A2/A1) (a) and increase in
groove depth (b).
appears. A summary of these two trends is presented in Fig. 4.17. Figure 4.17a shows how the band gap width changes asA2/A1 increases for each of the groove
depths. The band gap is measured in terms of wavelength for normal incidence and is in units of nanometres. Figure 4.17b examines the same band gaps but shows how the band gap increases as the groove depth increases.
If we compare the biharmonic structure to the single period case of the coupler grating in the first column of Fig. 4.16, a significant difference becomes apparent and the coupling to the first order Bragg band gap via this coupler grating is evident. As the band gap widens, the dispersion of the localised modes flattens. We are interested to see how far can this be pushed i.e. how flat the band can get and over what angular range. From the summaries presented in Fig. 4.17, it is apparent that increasing the grating depth has a significant effect on increasing the band gap size, while increasing the amplitude ratio has less effect. Consid- ering the two trends, further increasing the groove depth should achieve flatter bands.
One thing worth noting is as the band gap increases, the low frequency arm shifts to lower frequencies i.e. larger wavelengths. To compensate for this shift away from the near-infrared wavelength range, the period of the structure needs to be reduced. Through a variety of simulation experiments, it was observed that a period of 550 nm for deep grating shifts the localised modes of the low frequency arm of the band gap into the near-infrared wavelength range around 1.5 µm, as described.
The results for a biharmonic grating structure, with a Bragg-coupler ampli- tude ratio of 0.75, and for three different groove depths (600 nm, 800 nm and 1000 nm), are presented in Fig. 4.18. The plots only show the low frequency arm of the band gap. It is very clear that all three groove depths produce very flat dispersion bands, with the groove depth of 800 nm (Fig. 4.18b) producing the flattest band at 1.5 µm. Therefore, an incident wave couples to the same resonant wavelength regardless of the incident angle. This behaviour is quite extraordinary and not typical for periodic structures. For example, Fig. 4.15 shows how sensitive the resonant wavelength of a single period grating structure is to the angle of incidence. However, here we have designed a periodic structure
(d) (a) (b) (c) Groove Depth: 600 nm Groove Depth: 800 nm Groove Depth: 1000 nm Groove Depth: 800 nm (Note: Different Range)
Figure 4.18: Reflection dispersion plots for a biharmonic structure with amplitude ratio of 0.75 for three very deep groove depths; (a) 600 nm, (b) 800 nm and (c) 1000 nm. (d) Is a reflection dispersion diagram for the single period coupler grating structure with a groove depth of 800 nm and fill-factor of 50%. Note the change in the colour scale range.
that is completely angularly insensitive over the full half space of angles from normal to 90◦.
To prove that this effect is not purely a deep grating groove effect I have included the reflection dispersion diagram for the equivalent coupler grating with period 550 nm, groove depth 800 nm and fill-factor of 50%, Fig. 4.18d, for all of the biharmonic structures the fill-factor is 50%. The colour range has been changed to make it easier to see the resonant features. A flat, but very broad
resonant feature exists at 1.250 µm for the coupler grating. It shows how
weak the second order scattering can be when compared to the first order Bragg coupling of the biharmonic structure and proves the importance of including both.
Figure 4.19a shows the absorption spectra for the biharmonic structure of Fig. 4.18b, groove depth 800 nm and A2/A1 =0.75, for an incident angle range
of 0◦ to 45◦. Since there is no transmission through the grating, the absorption is the inverse of the reflection, A=1-R. The absorption spectra are plotted in
steps of 1◦ from normal incidence and are stacked up on top of each other,
to truly appreciate how flat the dispersion of the structure and how large the free spectral range is. The average linewidth is approximately 50 nm. In Fig. 4.19b the amplitude ratio is increased to 1 and the absorption is pushed up to
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0 20 40 60 80 100 Wavelength (µm) Absorption % 2 1 A /A = 0.75 θ= 0°to 45° 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0 20 40 60 80 100 Wavelength (µm) Absorption % A2/A1= 1.0 θ= 0°to 45°
(a)
(b)Figure 4.19: Absorption spectra for light incident on the same biharmonic structure
as in Fig. 4.18b with period 550 nm, groove depth 800 nm and A2/A1 =0.75 (a)
A2/A1 =1.0 (b). The incident angle ranges form normal to 45◦ in steps of 1◦. In both
plots the absorption spectra are plotted stacked on each other, note the rainbow of colours from blue to red.
almost 100%. Therefore, from Kirchhoff’s law of thermal radiation, this type of structure will make an excellent thermal emission device. When the structure is heated, the thermally excited SPP will couple to propagating photons via the deep biharmonic gratings and the emission wavelength over the entire half space will be the same, with an emissivity value of almost 1.
A key aspect to consider is the fabrication of such a binary biharmonic struc- ture, however. Unfortunately, I was not able to fabricate these structures during my thesis work. The two insets in Fig. 4.19 show just how high an aspect ratio the grating profile has for such a deep grating (800 nm) with a period of 550 nm. The etching of deep narrow grooves like this into a metal layer like gold is not easy. To relieve the issue it may not be required to etch such a deep grat- ing. Depending on the application, shallower groove depths still produce very flat dispersion bands with strong coupling, although, over a reduced angle range, see Fig. 4.18a. Using the RIE etching recipe presented in the earlier part of this chapter is an option, but, the redeposition of the etched gold is a serious problem. An alternative method to RIE etching process is focused ion beam milling.
Chapter 5
Coupled Cavity Array Thermal
Emitter
5.1
Introduction
In Chapter 3, we saw how a square hole array photonic crystal slab achieve narrow band thermal emission through resonant enhancement. The focus in Chapter 3 was completely on the Γ-point resonant modes, i.e. normal emission from the surface, with little attention paid to the off-normal (away from the Γ- point) behaviour of the structure. However, by examining the resonant modes in the band diagram of the square hole array slab (Fig. 3.1e (TE modes)), it is clear that, overall, they are not flat in k-space i.e. the resonant wavelength changes with the wave vector. Furthermore, in the opening section of the previous Chapter (Chapter 4), the off-normal spectral absorption behaviour for the same square hole array photonic crystal slab was presented (Fig. 4.1). The simulation results show that the resonant wavelength shifts and other resonant modes appear as the incident angle is increased from 0◦ (normal) to 80◦. This behaviour is not desirable as typically narrowband monochromatic sources are required, not sources where the emission wavelength changes as a function of angle.
In the previous chapter, two resonant structures with angularly independent resonant behaviour were introduced and designed; the metal-insulator-metal res- onator and the binary biharmonic structure. They both exhibited flat dispersion bands with no change in the resonant wavelength against incident angle. How- ever, because the emission material is gold, the resonances are broad and the devices are difficult to fabricate. In contrast, by using silicon as the emission material (Chapter 3), higher Q resonances are achieved and the devices are eas- ier to fabricate. Therefore, we require a structure which exhibits flat dispersion while maintaining narrow linewidth resonances and that can be fabricated from a material such as doped silicon.
In recent years, photonic crystal coupled cavity waveguides (CCWs) have gen- erated interest in the field of slow-light engineering [106]. Basically, the structure consists of a string of resonators, where light is coupled through the waveguide by evanescently coupling to each resonator. The system exhibits slow-light be- haviour but also demonstrates flat dispersion bands over large regions of their transmission spectrum. This type of structure shows promising behaviour.
Γ Χ Μ Γ 0 0.1 0.2 0.3 0.4 0.5 Wave vector Frequency (c/a) Γ Χ Μ TE modes TE band gap TE band gap Point defect Defect mode
Figure 5.1: Two dimensional photonic band diagram for a square array of holes in
silicon (n=3.5) with period a hole radius of 0.35a, TE modes only. The left inset
shows a cross-sectional view of the with a single hole, point defect. The right inset show the first Brillouin zone, with the irreducible Brillouin zone highlighted in orange. The yellow regions indicate the complete TE band gaps. A green horizontal line has been added to each band gap schematically to indicate the position of the single defect cavity mode.
ity array (CCA), which exhibit extremely flat dispersion bands. The structure consists of a typical photonic crystal of air holes in a dielectric slab like silicon. To create the cavity array, a hole is removed periodically across the 2D array of holes (e.g. every third hole removed). Each hole creates a defect and the light is trapped within the defect by the surrounding lattice, and therefore, forms a cavity. The coupled cavity mode typically falls within the band gap region of the photonic crystal lattice. CCAs have already demonstrated reasonable flat band behaviour for a square hole array photonic crystal lattice [107, 108], and have been used to achieve ultra-fast nanocavity photonic crystal lasing [109, 110].
Here, we further advance the design of the structure and achieve much flatter dispersion bands across the entire wave vector range. A triangular lattice of holes is used instead of the square array because it produces a larger band gap and so a larger free spectral range. 2D and 3D band diagrams are calculated for the structure. The number of holes between the defect cavities is altered to examine the coupling strength and to alter the dispersion relation. Finally, the material absorption is added by doping the silicon. The corresponding absorption spectra are calculated for normal incidence and, in steps of 10◦, upto an incident angle of 80◦ for both polarisations.