C-6.4 FACTOR DE REDUCCIÓN DE RESPUESTA

3.4.1 Simulation procedure

In addition to a narrow linewidth, hyperspectral imaging also requires that the photoresponse be tunable. The spectral properties of PC slabs are usually very sensitive to the geometry of the structure, so the period and hole radius are often altered for tuning. In this study, the period is held constant ata= 3.4 µm and the radius (r) is varied between 0.4 µm and 0.8 µm. In each design, all holes have the same size. The finite-domain structure and a TE-polarised plane-wave excitation were used for all simulations discussed here.

The measured Ex fields were also used to predict the spectral response of a QDIP integrated with that filter. There are many other factors that can influence the photoresponse, however this approximation may be used to select the most suitable GMRF designs. MOCVD-grown InGaAs/GaAs QDIPs have a response that peaks at about 6 µm and they have almost a 2 µm full-width at half-maximum (FWHM) [30]. In order to predict the response of the filtered detectors, the spectral response of a reference QDIP was modelled as a Gaussian curve centred at 6 µm with a FWHM of 2 µm. Modelling the GMRF as a passive filter means the response of an integrated device is the product of the reference photoresponse spectrum and the filter’s power transmittance spectrum.

Electromagnetic power is given by the time-average of the Poynting vector, which is equivalent to the real part of the complex Poynting vector, S, where

Re{S}=1_/2Re{E×H∗_{}. The intensity}

I is given by |Re{S}|. For a transverse-

electromagnetic plane wave, I = |E₀|2/(2η), where the impedance is η = 1/nqµ₀/₀, n is the refractive index, µ₀ is the free-space permeability and ₀ is the free-space permittivity [43]. E₀ is the magnitude of the electric field, which is essentially Ex for the TE simulations here. Hence, I ∝ |Ex|2 for TE plane-wave excitation, so|Ex|2 was used to predict the photoresponse of the integrated devices. The terms intensity and power will be used loosely in this section since the spectra are presented with arbitrary amplitude/intensity units.

to a loss of spectral resolution when the imaging target emits across the entire spectrum. This is akin to spectral crosstalk in wavelength-division multiplexed (WDM) communications. In these systems, crosstalk can be quantified using the overlap integral of the detection channel with an adjacent source channel, and then summing this quantity for all interfering source channels [44]. The present set of filters have a range of potential applications and are not limited to a specific source spectrum. To quantify the spectral redundancy between coupled GMRF designs, the overlapping area between pairs of photoresponse spectra was numerically calculated. This may be viewed as the worst-case crosstalk where the source spectrum is uniform across all bands of interest.

Finally, the quality factor (Q) was calculated for the reduced-domain periodic structure, with r = 0.60 µm. This was achieved by first calculating the Ex profile of the relevant resonant mode. Instead of using a pulsed excitation, an electromagnetic impulse was launched from inside the Ge slab. Subsequently, another impulse using the calculated mode profile was launched from inside the slab. A time monitor was placed at a point of low symmetry to measure the evolution of the total energy density in the slab. This is the sum of the E density and the H density at any time t. The Q value was then extracted from the exponential decay profile.

3.4.2 Results

Figure 3.7(a) shows the E field transmittance for r = 0.50 µm and r = 0.60 µm. These curves have been normalised to the peak from the latter design, which occurs at λ_p = 5.97 µm. In comparison, the r = 0.50 µm filter produced a peak E field at λ_p = 6.16 µm that is 10% smaller. These results suggest that r can indeed be used to tune the GMRFs, however some designs may perform better than others. Individual designs could potentially be optimised in future using numerical methods, such as particle swarm optimisation [45].

The predicted photoresponse spectra that would be measured from QDIPs integrated with these two filter designs are given in figure 3.7(b). The Gaussian spectrum representing a reference device is also shown. The FWHM for each filtered response spectrum is about 130 nm, which is less than 7% of the reference. The two peak wavelengths are also identical to the peak Ex wavelengths in figure 3.7(a). Clearly these filters can provide narrow photoresponse spectra with tunable λ_p and are therefore promising candidates for hyperspectral imaging systems. Some decrease in the peak photoresponse is observed. The integrated responsivity will

Figure 3.7: Effect of the hole radius (r) on the transmittance to QD depths, with a period of 3.4 µm. The normalised electric field (Ex) is given in (a), whereas (b) shows the predicted photoresponse for each design. In (b), a normalised Gaussian spectrum is assumed for the unfiltered QDIP response and the pink shading indicates the overlap area between the two filtered designs.

also suffer with narrow passbands but this is a small price to pay for the addition of hyperspectral functionality.

The pink shading in figure 3.7(b) indicates the overlap area between the two spectra. This represents a loss of spectral separation between these designs and should be minimised. There are several factors that contribute to this overlap. Firstly, peaks that are close in λ_p will obviously have more area in common. Secondly, the short-wavelength shoulder of ther = 0.50 µm peak is entirely within the r = 0.60 µm band. These shoulders were found to be smaller under TM excitation, so the actual size may be an average of both polarisations and this may slightly decrease the overlap area. Finally, the finite background transmittance contributes to a broad background that is common to all of these designs. This is a consequence of the asymmetrical GMRF design in which the CaF2 cladding

and GaAs substrate have different refractive indices to the air and the Ge slab, respectively. This issue will be discussed in section3.8.

A systematic study using 17 different r values was performed and the resulting spectra are displayed in figure 3.8, with the colour scales representing arbitrary units in electric field and photoresponse. The Ex spectra for each hole size are shown in figure3.8(a), which were normalised to the source pulse in order to extract the transmittance, but have not been normalised to any peak. The predicted photoresponse spectra in figure 3.8(b) were calculated as for figure 3.7(b), but then normalised to the strongest peak (r = 0.60 µm). For r > 0.7 µm, some of the photoresponse peaks are red-shifted by about 10 nm from the respective E

Figure 3.8: Tunability of the transmittance spectrum with varying hole radius (r) and a period ofa= 3.4 µm. The electric field (arbitrary units) is shown in (a) and the predicted photoresponse (normalised arbitrary units) is shown in (b), forr ∈[0.4,0.8] µm.

peaks, however in other cases the peak positions are identical. There is a clear relationship between hole radius and this peak location. The peak wavelength of the predicted photoresponse can be approximated as λ_p = −1.0(d) + 7.1, where d = 2r is the diameter of the holes. The Pearson correlation coefficient for this linear regression is −0.9988, indicating a strong relationship. This effect can be explained by the fact that the average refractive index of the Ge slab decreases as the air holes become larger. For a triangular lattice, this average index is √_g = [_H+ 2πr2(_L−_H)/(√3a2)]1/2, where

_H = 16 and _L = 1 are the relative permittivities in the Ge and the holes, respectively. As the average index decreases, the photonic bands and hence the guided resonances are moved to higher frequencies [26]. The slight bowing of the trend in figure 3.8 is also reflected in a plot of _g against r (not shown), however for both cases a linear approximation is more than adequate.

It has been reported that decreasing the hole radius will produce sharper resonances with longer lifetimes [26]. This is explained by the fact that GMRs will approach true guided modes asr→0. The FWHM of theEx peaks in figure3.8(a)decreases from 0.37 µm to 0.22 µm as r is decreased from 0.80 µm to 0.525 µm. As the holes are further shrunk, however, the linewidth rapidly increases to 0.41 µm atr = 0.40 µm (not shown). The short-wavelength shoulder previously discussed becomes significant for these filters and leads to larger FWHM values. The peak height also falls off sharply for very small holes. Therefore these designs may require some improvement (for example by optimisinga) if narrowband QDIPs are required with λ_p >6.2 µm.

Figure 3.9: Overlapping area (arbitrary units) between the predicted photoresponse design pairs. Each design is expressed in terms of hole radius (r). In (a), each element is normalised to the largest peak area (self-overlapping). In (b), each column is normalised to the self-overlap area of the selected band.

The overlapping area between the predicted photoresponse of two devices was shown in figure 3.7(b). The corresponding overlap was numerically calculated between all combinations of the 17 designs. This is shown in figure 3.9(a), where areas have been normalised to the largest value. All of the largest areas fall along the diagonal, since these are just the individual peak areas and they represent the self-overlap of each design. Clearly, the peaks with the largest area occur for r ≈ 0.6 µm and r ≈ 0.75 µm, so detectors coupled with these designs should have the highest responsivities. On the other hand, the peak area falls off sharply for small r as these are the long-λ_p designs where the peak heights are lower. These are also the peaks where the FWHM becomes larger, and this can lead to significant spectral overlap. This is highlighted in figure 3.9(b), where each column of overlap areas has been normalised to the peak area on the diagonal. For multicolour applications where only a few bands are needed, one can use these figures to choose the GMRF designs. If one design is selected according to λ_p, then the choice of other designs should consider the amount of spectral overlap. For example, most filters overlap with a significant proportion of the r = 0.4 µm design and so this device will not provide much additional spectral information. In contrast,r= 0.5 µm could be paired withr >0.7 µm filters, as these devices will be spectrally separate. When any two bands are required (and the specific application does not dictate λ_p), then r = 0.60 µm and r = 0.75 µm are clearly preferable, since their individual peak areas (responsivity) are large and the amount of overlap is minimal. In hyperspectral systems, some overlap might be tolerated in order to provide contiguous spectral information.

Figure 3.10: To determine the quality factor (Q) value of 44 for ther= 0.60 µm structure, the profile of the resonant mode was determined as shown in (a). An impulse with this mode profile was launched inside the slab and the total energy density (arbitrary units) was monitored, which is shown in (b). The inset shows the region selected to extractQ and the exponential nature of the decay is emphasised with a logarithmic scale.

Given the trends in peak height and resonance linewidths, the designs with r ≈

0.6 µm would be expected to give the highest quality factors (Q). Q relates to the amount of energy stored in a resonant mode, U(t), and the rate of energy loss. This can be approximated with the frequency spectrum by the relation Q = ν_p/∆ν, where ν_p is the peak frequency of the resonance and ∆ν is the FWHM linewidth. For isolated modes, the energy decays exponentially in the form U(t) = U₀exp(−αt), where α is the rate of decay. In this case, Q = 2πν_p/α [38]. This relationship was used to calculate Q for the r= 0.60 µm design.

Figure 3.10(a)shows the profile of the resonant Ex mode around 6 µm. Although the monitors at QD depths measured an Ex peak of 5.969 µm for r = 0.60 µm, choosing the mode profile for 5.995 µm resulted in a purely exponential decay. The total energy density is shown in figure 3.10(b), and the inset emphasises the exponential decay with a logarithmic U axis. Note that time is measured in µm because FullWAVETM defines time in terms of ct, where c ≈ 3×1014 µm/s is

the speed of light in vacuum. In other words, light will travel 1 µm in free space over each unit of time. The range of times shown in the inset of figure 3.10(b)

was used to extract α = 0.0237 µm−1. Since ν_p = 1/5.995, then Q = 44. In fact, the frequency spectrum (not shown) gives an estimate of Q ≈ 43, which is

in very good agreement. Using this graphical approximation, two other resonances observed at 3.74 µm and 4.49 µm were estimated to have quality factors of 67 and 111, respectively. Note that for some PC slabs, ultra-high Q values in the order of 106 _{have been reported, and these are practically limited by fabrication}

small modal volume is desirable, for example in PC lasers [47]. GMRs often have lowQ values and have other applications, such as increasing light extraction from light-emitting diodes [48].