A novel integrated optic photodetector with resonant length is theoretically proposed for detection at 1.55 μm. The structure consists of substrate/waveguide/phase matching layer (low index)/absorbing semiconductor layer/air. A general mode expansion and propagation technique based on supermode analysis is used to study the propagation of modes in the device structure. Periodic coupling between the guided mode and the mode supported by the absorbing layer takes place and varies with the length of the detector. We show that resonant coupling between the two super modes takes place for the optimized thickness of the phase matching layer and the absorbing layer at a particular length of the detector which is considered the detector length. An attenuation exceeding 170 dB could be obtained for a detector length of 13 μm for TM polarization. The proposed photodetector exhibits a 3dB bandwidth exceeding 300 GHz and has capacitance of only 0.01 pF. Similar results have been obtained for TE polarization with different thickness of the absorbing layer.
1.Introduction
Photodetectors are essential elements of the receivers. Efficient photodetectors must have high sensitivity, fast response (short response time), low power consumption, high quantum efficiency and high bandwidth to handle required data rates. The photodetector should be compact so that it can be used in the integrated optics applications. To meet these requirements, it should be low cost and compatible with the existing fabrication techniques. At present, a variety of photodetectors is in existence including photomultipliers, phototransistors, photodiodes and waveguide photodetectors [1–4]. Although photomultipliers have very high gain and low noise they are not suitable for optical waveguide systems due to their large size and high voltage requirements. Phototransistors and photodiodes are exclusively suitable for fiber optic systems and cannot be incorporated into a waveguide structure. Integration of photodetectors with optical waveguides is desired because the processing of optical signals requires conversion to electrical signals. The photodetector can also be integrated with planar waveguide sensor structures.
Optical energy from the waveguide can be coupled to the photodetector either as side illumination or evanescently [5, 6]. Evanescent coupling to the photodetectors is preferred as it offers low coupling losses. Recently, many such photodetectors have been studied [6–16]. A monolithic integrated waveguide photodetector with optical isolation between the detecting layer and the contact with responsivity of 0.55 A W−1 has been reported [7]. Silicon nitride waveguide integrated silicon p-i-n photodetectors on SOI platform has been proposed [8]. Waveguide photodetectors based on lateral Si-Ge-Si heterojunction have been demonstrated at 1.55 μm [9]. Photodetectors based on Er/O doped Si waveguides have been produced with a 3-dB bandwidth of 30 KHz [10]. Although compatible with already mature and standard CMOS technology, the germanium and silicon based photodetectors have serious limitation of higher (than the commonly used III-V photodetectors) dark current densities. Photodetectors based on III-V quantum wells on the Si waveguides have been reported [11]. A uni-traveling carrier photodetector fabricated in InP based photonic membrane bonded on a Si wafer has been studied [12]. InGaAsP/InP segmented waveguide photodetector based on directional coupler utilizing the matching of imaginary part of the propagation constants of the modes has also been demonstrated [13]. InGaAs/GaAs multi-quantum well p-i-n nano-ridge waveguide based photodetector has been presented with responsivity of 0.65 A W−1 [14]. Optical components based on the III-V semiconductors (and their alloys), due to their better optical and electronic properties have been successfully realized and commercialized and are also potential candidates for designing photodetectors in communication wavelength region. The photodetectors mentioned above could be designed to get quantum efficiency ranging from 0.35 to 0.90, a bandwidth upto 100 GHz and length between 10 μm and 400 μm. Irrespective of the materials being used to design the device, the light coupling from the lossless waveguide to the absorbing layer takes place through evanescent coupling. The absorbing layer (or structure) placed on the lossless waveguide acts as a loaded waveguide. The detector thickness can be varied to improve the absorption. Alternately, the length of the detector can also be varied to enhance absorption. Direct loading of the absorbing layer on the waveguide, often results in phase mismatch (or impedance mismatch) between the guided mode and the optical mode in the detector layer, thereby reducing the coupling efficiency.
We propose a novel structure to overcome the phase mismatching between the guided mode and the mode supported by the absorbing layer. It is shown that low index layer inserted between the waveguide and the detecting layer can provide perfect phase matching due to which optical tunneling of the mode power takes place from the waveguide to the detecting layer. The layer thickness can we optimized to design short length and low capacitance photodetector therefore improving the quantum efficiency and the speed. Photodetectors for both TM and TE polarizations have been designed.
2.Proposed structure and optimization
The proposed structure is shown in figure 1.
The detecting multilayer section consists of an AlAs substrate (ns = 2.9), Ga0.6Al0.4As waveguide (nf=3.15, tf=1.0 μm), low index AlAs matching layer (nm = 2.9, tm), absorbing detector layer of GaInAs (na = 3.53–0.0764i, ta) and air clad (nc = 1.0). Where nj , tj respectively are the refractive index and thickness of the jth layer. The substrate and the cladding are semi-infinite in extent. The substrate and the waveguide are common to all the three sections.The operating wavelength (λ) is 1.55 μm, which, in general, is used for telecommunication operations. As shown in figure 1, the photodetector consists of three sections, the input waveguide, the multilayer detecting region and the output waveguide. The input and output waveguide sections are similar and consist of basic three layer structure. The central detecting region consists of a waveguide layer and the absorbing layer of GaInAs separated by a low index matching layer. The mode eigenvalue of the three sections have been obtained by solving a general eigenvalue equation of a multilayer structure. The multilayer eigenvalue equation defining the multilayer structure is solved using the robust argument principle method followed by root polishing technique [17]. The propagation of light through all the sections of the photodetector structure was carried out using the mode expansion and propagation method [18]. The input waveguide provides power to the photodetector section which excites the two modes supported by it. The absorption losses, in dB, in the detecting section can be expressed as [19]
where 
and 
are the coupling coefficients at the input z = 0 and output, z = L, positions of the photodetector. 
are the propagation constants of the two eigenmodes of the photodetector section.
3.Design and analysis
The photodetector section can be considered as the structure consisting of two waveguides, a lossless waveguide with layer index nf and another an absorbing layer with complex index and na +jka where ka = λα/4π, α is the material absorption coefficient of the layer. The light propagating in the waveguide (entering from the input section) evanescently couples to the absorbing detector layer. In general, the coupling efficiency is not very high due to phase mismatching between the two modes of the photodetector section. In such photodetectors the absorption is dependent on the length of the photodetector and results in low quantum efficiency. We introduce low index dielectric layer between the waveguide and the absorbing photodetector layer. For optimized layer thicknesses, the optical power in waveguide tunnels through the low index matching layer to the absorbing layer. The thickness of the absorbing layer is chosen to support only one mode. When the index matching layer thickness is large, the modes supported by the photodetector section do not interact with each other and we can consider them as independent modes supported by the two to parallel waveguides. However as we reduce the thickness of the low index matching layers, the two modes start interacting with each other and their effective mode indices change. With the decrease in matching layer thickness, index difference between the two guided modes also changes. Even for large matching layer thicknesses the two modes can interact with each other if the indices of the modes supported by the waveguide and the absorbing layer are made equal and the structure can be studied using coupled mode theory [2, 18]. This often results in a very long coupling region for the power exchange to takes place. In this case modes of the waveguide are just perturbed by the presence of each other. For designing a compact photodetector, we will use the super mode analysis and assume that two modes are supported by a multilayer waveguide structure rather than consisting of two separate waveguides. This is possible when the matching layer thickness is very small (or even zero). The two modes now not only interact with each other but also interfere. The matching layer thickness is so adjusted that a constructive interference takes place between the two eigenmodes supported by the photodetector section, thereby providing a very strong absorption (coupling) of the light by the absorbing layer. We have designed the waveguide photodetectors for both TE and TM polarizations.
3.1.TM mode photodetector
Let us first consider the photodetector for the TM mode. The absorption and mode effective indices of the waveguide can be controlled by changing the thicknesses of the matching layer and the absorbing layer. The optimized thicknesses of various layers are tf=1 μm, tm=0.09 μm and ta=0.38 μm. The 1μm thick waveguide supports a well guided single mode and is kept fixed for all the sections. tm and ta are adjusted to produce maximum absorption in the absorbing layer. This happens when the guided mode and the mode supported by the absorbing layer are in same phase. The low index matching layer provides perfect phase matching between the modes and constructive interference takes place. The energy is completely absorbed in the absorbing layer. The length of the detector layer (resonant length) corresponds to a field localized in it and it is called coupling length or half beat length and is represented by L=λ/2(neff1 —neff2 ), neff1 and neff2 being the effective mode indices of the two supermodes in the PD. In this case L=13.0 μm. The absorption of the TM mode power, as a function of the photodetector length, is shown in figure 2.
There is a resonant absorption (171 dB) peak at L=13 μm which determines the photodetector length. There is a periodic coupling of power between the two eigenmodes supported by photodetector, although resonant peak appears only at 13 μm. As the modes travel down the photodetector section, their phases change with distance. The amplitudes of the eigenmodes are also complex and contribute to their phase accumulation. Since the amplitude phases are not periodic, the phase matching between the two modes takes place only at z=L and the resonant attenuation peak appears only once. The strong power transfer can also be explained on the basis of field plots for the two eigenmodes at L=13 μm as shown in figure 3.
The fields of the two hybrid modes are out of phase in the waveguide region and in phase in the absorption layer. The energy is thus transferred to the absorbing layer, where it is completely absorbed due to mode field interference. The transfer of energy to the absorption layer and its absorption can be obtained by superposition of the two modes. Figure 4 shows the composite mode field profile resulting from the superposition. Propagation of the mode energy along z-direction is shown on x–z plane. x (vertical) direction defines various layer thicknesses of the detector section. The color bar on the left side of the guiding structure represents the relative intensity scaling.
At z = 0 μm, most of the mode energy lies with the waveguide and gradually transfers to the absorbing layer. At resonant length, L=13 μm, the energy is almost completely absorbed by the lossy layer.
The quantum efficiency, 
of the integrated optic photodetector can be expressed as [2]
For 
The capacitance of the PD can be expressed as
where, ε is the permittivity, A is the area of cross section and d is the thickness of the capacitor (here, tf+tm+ta).
If we consider the detector width equal to 10 μm, then for L=13 μm, we get C=0.0094 pF.
The bandwidth (BW) of the detector is 
For a load resistance, 
we get 
The responsivity of the photodetector is [1]
is electron charge, 
is the Planck's constant and 
is the velocity of light in vacuum.
At λ=1.55 μm, 
The wavelength dependence of the attenuation of the photodetector is also shown in figure 5. Only the complex refractive index of the absorbing layer is considered to be wavelength dependent [20].
The attenuation of the photodetector is much higher even away from the resonant wavelength as compared to previously reported ones. This makes it usable for a wider wavelength region.
3.2.TE mode photodetector
The photodetector is also designed for the TE polarization. The operating wavelength is again 1.55 μm. Since mode profile of the TE mode is different from that of the TM mode, the thicknesses of the matching layer and the absorbing layer are again optimized to get resonant attenuation peak. It appears at tm=0.09 μm and ta=0.30 μm. There is change in only the absorbing layer thickness. The attenuation plot is shown in the figure 6.
The resonant peak appears at L=14 μm, corresponding to an attenuation of 173 dB. The optimized structure results in similar values of 
(0.01 pF) and BW (318 GHz). The energy transfer from the waveguide to the absorbing layer is shown in the figure 7.
At resonant length, the energy is completely depleted from the waveguide as it appears in the figure 7.
4.Conclusions
In summary, we theoretically propose a resonant length integrated optic photodetectors for both the polarizations (TE and TM). The absorbing (detector) layer is separated from the waveguide by a low index dielectric layer. The thickness of the absorbing layer is chosen to support a single guided mode. We show that the low index matching layer helps the guided mode to resonantly couple (tunnel) to the mode supported by the absorber by matching the phases of the two modes. The large attenuation in the device is a result of constructive interference of the two modes at the half beat length. Because of the resonant coupling such a structure provides a large attenuation, small detector length for a thin absorbing layer and enhances the bandwidth by lowering the device capacitance. We show that high responsivity (quantum efficiency) and large bandwidth can be achieved simultaneously with a resonant length photodetector. The obtained results show great potential of the proposed structure in designing high quantum efficiency and short length photodetectors without taking too thick absorbent layer. The design procedure can also be applied to other class of photodetectors such as germanium on silicon or silicon on III-V materials or silicon based photonic integrated circuits. The proposed photodetector can also be attractive for sensing and high speed data communication applications.
Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).
