Chapter 4 The characteristic of cavity modes in the
4.2 Cavity thermal emitters with randomly distributed hole array and
size to the cavity modes
CMs exhibited better performance than SPPs in the emission spectra of a PTE. It can be used to fabricate better thermal emitters if Bragg scattered CMs and SPPs
modes can be filtered out. In this section, CTEs with randomly distributed hole array (RDHA) are proposed to eliminate these unwanted modes which are associated with surface periodic structures. Beside, the influence of hole size to the emission peaks and reflection spectra is investigated.
4.2.1 Experiments
The schematic cross section and top view of CTEs with RDHA are shown in Figs.
4.5 (a) and (b), respectively. The fabrication processes are the same as described in previous section, eleven samples D to N were prepared for experiments and their structure parameters are summarized in Table 4.3. The reflection spectra of sample N is measured along ΓK direction with angle ϕ normal to the surface as defined in Fig.
4.1 (b). For samples D to F, there are no holes perforated in the surface metals so that
(a) (b)
Fig. 4.5 (a) The side view and (b) top view of CTEs with RDHA.
Table 4.3 The structure parameters of samples with randomly distributed hole array, d denotes the diameters of holes and h denotes the thickness of to silver film.
Sample d (μm) h (nm) DH* (%)
D 0 12.5 0
E 0 15 0
F 0 20 0
G 1 100 6.2
H 1.5 100 9
I 2.5 100 9
J 3.5 100 9
K 4.5 100 9
L 2 100 6.2
M 2 100 9
N** 2 100 40.3
*DH: Density of holes: total hole area over total area of pattern
**Sample N is a CTE with hole array arranged in hexagonal lattice instead of RDHA in order to offer maximum density of holes for comparison. The period of hole array for sample N is a=3μm.
there is no need of lithography and lift-off in the fabrication processes. Simulations [45, 51] suggest that once the thicknesses h of top Ag layers are in the order of skin depth (~20nm), intrinsic CMs can be excited in the cavity and radiate to the far-field by leaking through surface thin film and vice versa. These kind of structures can be viewed as one kind of CTEs with zero hole size.
The masks of RDHA used for lithography were designed by the computer program. A unit basis of 500 μm x500 μm RDHA was generated and then spanned into the size of 1cmx1cm by periodic mapping for all samples. The nearest edge-to-edge distance allowed for holes is 1 μm . The distribution of the nearest center-to-center distance for all holes are calculated and recorded in the step of mask-design by counting the nearest center-to-center distance from one hole to all other holes within the area of unit basis. The distributions of numbers of holes as a function of the nearest center-to-center distance are chosen to be as decentralized as possible without a peak at certain wavelengths. This is because large number of holes with common nearest center-to-center distance makes the distribution of the lattice momentum in the reciprocal lattice concentrate in some point (GJJG1
,GJJJG2
). These high
weight of G JJG1
and GJJJG2
would contribute into the Eqs. (4.4) and (4.5) even if surface holes are distributed randomly without period [70].
All patterns of RDHA are checked in the extraordinary transmission. The
fabrication and measurement methods are the same as those described in Sec 3.1. Fig.
4.6 compares the distribution of the nearest center-to-center distance of holes with the transmission of RDHA in normal direction for sample I. The relation of transmission maximum to the distribution maximum follow the following equation
eff
where λ is the wavelength of the maximum transmission. T a is the nearest eff
center-to-center distance of all holes whose weight are the largest. n=3.45 is the refraction index of the Si substrate. Eq. (4.15) is the same as Eq. (4.12) so that a eff can be viewed as effective period of hole array. However, due to highly decentralized distribution of the nearest center-to-center distance among holes, the transmission is very weak and there is no apparent EOT observed in the spectra. Not only the RDHA patterns used for sample I but also other RDHA patterns had been checked in the transmission experiments to make sure that all RDHAs are random enough without effective period for all samples. The maximum density of holes for sample G is 6.2%
which can not be elevated further without reducing the minimum distance between neighboring holes.
Fig. 4.6 The relation between transmission and the distribution of the nearest center-to-center distance of holes for sample I.
4.2.2 Results and discussion
For CTEs with RDHA, the SPPs modes can not be excited due to lack of periodicity coupling as discussed in Sec 2.2. RDHA can be viewed as a periodic structure with period of infinite; substituting Eqs. (4.6) and (4.7) into Eq. (4.4) with
a= ∞ yields
For zero order measurement (ϕ =0o), the wavelengths of emission peaks in vacuum can be obtained by solving Eq. (4.16) withk =0 //
ox SiO2
CM
2 t n λ =
m
× ×
(for CMs) (4.17)
Eqs. (4.16) and (4.17) are identical with Eqs. (4.9) and (4.11) in the condition of k= =0A , this means that only intrinsic CMs without SPPs and Bragg scattered CMs
can be excited in the cavity. The emission spectra should be pure and narrow-band with the intrinsic CMs only. The theoretical dispersion curves according to Eq. (4.16) are shown in Fig. 4.7. The SiO2 thickness used in the calculation is 2 μm.
Fig. 4.7 Theoretical dispersion curves of CM for CTEs with RDHA. tox=2 μm
Figs. 4.8 (a)-(c) show the experimental dispersion relation of reflection spectra with measuring angle ϕ from 12o to 65o for samples D to F where no holes exist in the surface silver film. Figs. 4.8 (d)-(f) show the comparison of theoretical and experimental results for samples D to F, respectively. The blue dashed lines are the calculated results of Eq. (4.9). The symbol (0,0,m) CM denoted in the figures represent the CMs whose k= =0A and m satisfies Eq. (4.9). The spectra are much cleaner compared with CTEs with periodic hole array as shown in Figs. 4.3 and 4.4.
For sample D shown in Fig. 4.8(a), the 12.5nm thick surface thin film is not thick enough so that large leakage of light into the cavity occurs no matter whether the incident light can couple to cavity modes or not. The cavity modes can propagate in the waveguide and leak out of cavity through surface thin film and results in a bright band in Fig. 4.8 (a). The leakage of light out of cavity also increases the bandwidth of resonance modes. For sample E shown in Fig. 4.8 (b), the bandwidth of CMs become much smaller and the reflections are still very low. For sample F shown in Fig. 4.8(c), the thickness of top thin film is 20 nm which is too thick to enable the light leaking from surface into the cavity if it is not a cavity mode, therefore, the reflection coefficient is almost one; however, for light coupled to cavity mode, the reflection becomes weak and dark line appears, the reflections of (0,0,1) CM and (0,0,2) CM for sample F are higher than the reflections of (0,0,1) CM and (0,0,2) CM for sample E
Fig. 4.8 The dispersion relation of reflection spectra for sample (a) D, (b) E, (c) F and their modes analysis for sample (d) D, (e) E and (f) F. The blue dashed lines are the calculated results of Eq. (4.9).
due to lower coupling efficiency of light to CMs whose surface silver film is thicker.
Figs. 4.9 (a)-(c) show the dispersion relation of reflection spectra for samples G to I, respectively, not only the upward curved intrinsic CMs but also the horizontal dark lines and bright lines are observed for all spectra as denotes as (0,0,m) LCM which indicate the characteristic that horizontal (0,0,m) LCMs always intersect with (0,0,m) CMs at k//=0. The horizontal lines act as the localized cavity modes (LCMs) independent on the direction of incident light. Beside, a broad band horizontal dark lines below (0,0,1) LCM is observed either for Sample I denoted as FP-hole,
Fig. 4.9 The dispersion relation of reflection spectra of samples (a) G, (b) H and (c) I.
Actually, these LCMs are the localized standing waves resonated in the cavity in the direction normal to samples in the cutoff frequency
SiO2
ox
ω mπ
n =
c t (4.18)
The incident lights are scattered by the holes once the wavelengths of LCMs are smaller than three times of hole diameters. Fig. 4.10 shows the theoretical dispersion curves of CMs and LCMs according to the Eq. (4.16) and (4.18). Two curves with common m value would interest at k//=0. Consider the dispersion relation of reflection spectra for samples G to I as shown in Figs. 4.9 (a)-(c), respectively. For sample G, the hole diameter is 1μm, (0,0,1) LCM does not appear since the theoretical wavelength of (0,0,1) LCM is 5.4μm which is three times larger than the hole diameter. The hole diameter is too small to scatter the incident light to form the (0,0,1) LCM. For (0,0,2) LCM, the wavelength of (0,0,2) LCM is 2.87μm which is smaller than the three times of hole diameter, the holes scatter incident lights so that forms the (0,0,2) LCM.
For sample H, the hole diameter is 1.5 μm, very weak (0,0,1) LCM seems to appear in the theoretical position, The theoretical wavelength of (0,0,1) LCM is 5 μm which is larger than three times of hole diameter but the difference is small, very weak (0,0,1) LCM are formed due to low scattering efficiency of holes. For sample H, the hole diameter is 2.5 μm, both (0,0,1) and (0,0,2) LCM appear clearly since the wavelengths of (0,0,1) and (0,0,2) LCM are 5μm and 2.87 μm , respectively, which
Fig. 4.10 The theoretical dispersion curves of CMs and LCMs.
are all smaller than three times of hole diameter.
It is expected that when the hole diameter is small (1μm), only the incident light coupled to the cavity modes can have extraordinary transmission through top metal film into the cavity, this leads to a weak reflection and dark lines. However, when the hole diameter becomes large (2.5 μm), the transmission of incident light with different wavelengths also becomes significant, only those light coupled to cavity
mode can propagate in the cavity, all others are scattered away. These cavity modes will re-emit from the holes to the far-field and enhance the reflection spectra (bright lines).
Consider the dispersion relation of reflection spectra for samples J and K whose diameters are 3.5 and 4.5 μm, respectively as shown in Fig. 4.11 (a) and (b), respectively. For larger hole size and shorter wavelengths, the LCMs gradually disappear, this is because the leakage of light resonated in the cavity through surface hole array becomes too large to be ignored. Highly leakage of light for larger hole size breaks the resonance condition in the direction normal to the surface of samples. This makes the dispersion relation of reflection spectra for large holes gradually approach to the dispersion relation of reflection spectra for sample D whose leakage of light in the cavity is larger either, as shown in Fig. 4.8 (a).
Fig. 4.11 The dispersion relation of reflection spectra of samples (a) J and (b) K.
The wavelengths of CM and LCMs are independent of hole diameter. However, the wavelengths of the other horizontal broad-band dark modes denoted as FP-hole in Fig. 4.9 (c) and Fig. 4.11(a) and (b) are not. They are the Fabry-Perot hole shape resonance (FP-hole) modes resonated in the edges of the holes and whose wavelengths are linear dependent with the hole diameter. Fig. 4.12 shows the measured wavelength of FP-hole modes to the hole diameter with the incident angle
=12o
ϕ for samples I to K and another CTE sample with d=tox=2 μm. The linear fit
curve of measured points is
λ=4.14+0.64d (4.19)
Fig. 4.12 The relation of the wavelength of FP-hole modes to the hole diameter
Fig. 4.13 shows the reflection spectra at ϕ=12o for samples L to N whose diameters of holes are fixed at 2 μm but the densities of total hole area are different.
The holes of sample N are arranged in hexagonal lattice instead of RDHA in order to offer the maximum values of density of total hole area to 40% for wide range comparison. Without considering the SPPs and Bragg scattered CMs of sample N, it can be seen that the wavelengths of dark modes (0,0,1) and (0,0,2) CMs are almost the same except a tiny red-shift for larger densities of total hole area. Higher densities of holes only lower the reflection intensity of modes.
Fig. 4.13 The reflection spectra of sample L to N with the fixed incident and reflective angles 12ϕ= o and the fixed diameters of holes d=2μm.
Finally, the emission spectra of samples D to J were investigated; none of CTEs with top thin silver films without holes (samples D to F) 0can offer good emission spectra in the temperature over 140o C. Fig. 4.14(a) shows the emission spectrum of sample E. Once the temperature was elevated from 80o to 100o C, the quality of surface thin film degenerates so that the background thermal radiation of SiO2 phonon vibration modes around 10 μm [53] can not be suppressed the surface thin films are too thin to offer the good thermal stability in high-temperature operation. For CTEs with RDHA, the emission spectrum of sample G is too weak to be ignored due to low density of holes. For the emission spectrum of sample H at 300oC as shown in Figs.
4.14 (b), the wavelength, FWHM, Δλ/λ and Q factor of (0,0,1) CM is 5.51 μm, 0.17, 0.03, 33, respectively, which are as good as the characteristics of (0,0,1) CM of sample B as summarized in Table 4.2. However, the intensity of (0,0,1) CM is very weak due to low density of total hole area by comparing Fig. 4.14 (b) with Fig. 4.2 (b).
For the emission spectra of samples I and J at 300oC as shown in the Fig. 4.14 (c) and (d), respectively, larger hole size broaden the bandwidth of resonance modes so that the difference between emissions spectra and background blackbody radiation gradually smears out. It should be noted that the maximum density of total hole area can not be enlarged too much in order to maintain the random degree of holes. Low output intensity is the main drawback of CTEs with RDHA.
(a)
(b)
(c)
(d)
Fig. 4.14 The emission spectra of samples (a) E, (b) H, (c) I and (d) J in the normal direction (ϕ=0 )o .
In conclusions, CTEs with RDHA had been realized successfully, by analyzing the hole size effect on the resonant modes in the dispersion relation of reflection spectra, it is found that larger scattering of light through larger surface holes array would form the LCMs and FP-hole. When the hole diameter is small, only the incident light coupled to the CMs and LCMs can have extraordinary transmission through top metal film into the cavity, this leads to a weak reflection and dark lines.
When the hole diameter becomes large, the transmission of incident light with different wavelengths becomes significant, only those light coupled to CMs and LCMs can propagate or resonate in the cavity and re-emit from the holes to the far-field so that changes the reflection spectra from dark lines into white lines. When the hole diameter is larger than or equal to 2.5 μm, the FP-hole modes appear and the wavelengths of FP-hole modes are linear dependent on the hole size. Finally, the emission spectra of CTEs with RDHA are pure and narrow-band if the hole size is small. However, their output intensities are very weak due to low density of total hole area.