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Radiation-hardened silicon photonic passive devices on a 3 µm waveguide platform under gamma and proton irradiation

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Abstract

Silicon photonics is considered to be an ideal solution as optical interconnect in radiation environments. Our previous study has demonstrated experimentally that radiation responses of device are related to waveguide size, and devices with thick top silicon waveguide layers are expected to be less sensitive to irradiation. Here, we design radiation-resistant arrayed waveguide gratings and Mach-Zehnder interferometers based on silicon-on-insulator with 3 µm-thick silicon optical waveguide platform. The devices are exposed to 60Co γ-ray irradiation up to 41 Mrad(Si) and 170-keV proton irradiation with total fluences from 1×1013 to 1×1016 p/cm2 to evaluate performance after irradiation. The results show that these devices can function well and have potential application in harsh radiation environments.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

High-energy irradiation can cause degradation and even failure of semiconductor devices by introducing total ionizing dose (TID) and displacement damage (DD) effects. During space missions, spacecraft are exposed to high-energy particles and cosmic rays [1]. Nuclear reactors and particle accelerators in terrestrial environments also lead to damage in sensitive materials and devices. Silicon photonic integrated technologies have unique advantages as optical interconnects and data transmission systems for applications in harsh environments due to lightweight, compact, and low-power consumption [2]. In passive optical interconnect systems, optical filters are widely used for applications in wavelength division multiplexing (WDM). The most common optical components used as filters include arrayed waveguide gratings (AWGs) and Mach-Zehnder interferometers (MZIs) [3]. MZIs and AWGs are typical commercially available building blocks that are used as compact WDMs [46] due to their high index contrast. These optical components work based on interfering light beams, and they are vulnerable to high energy irradiation.

Radiation effects of devices in different technology platform have been investigated recently, principally based on silicon oxynitride [7], silicon nitride [8,9], amorphous silicon [8,10], silicon-on-insulator (SOI) [1114], silicon carbide [15], InGaAsP/InP [16], silica-on-silicon (SoS) [14], polymers [10,17] and LiNbO3 [18]. As reported in [7], silicon oxynitride MZIs experience an increase of 10−2 in the refractive index of SiON and SiO2 after 2.3 MeV α-irradiation to a fluence of 1015 cm-2, and the modal effective refractive index changes by 10−3. Wavelengths of SOI MZIs and micro ring resonators (MRRs) are found to shift and the modal index changes by ∼10−3 after exposure to neutron irradiation at a fluence of 1012 n/cm2 [13]. To qualify devices for space applications, the SoS 1-by-40 AWG demultiplexers and SOI MRRs are exposed to 60Co γ-rays to a total dose of 300 krad(Si). In these tests, the wavelengths of SoS AWGs are found to shift ∼0.03 pm/krad(Si). For SOI MRRs, blue shift in wavelengths of ∼0.33 pm/krad(Si) have been observed [14]. In addition, G. Brunetti et al. have investigated the effects of 60Co γ-ray irradiation on a high Q-factor InGaAsP/InP ring resonator [16]. The resonance peak exhibits a red shift; variations of 13% in Q factor and 4% in extinction ratio (ER) are measured at ∼ 320 krad(Si).

Modal effective refractive indices are affected by radiation damage, thereby degrading properties of devices. Ionization damage can produce trapped charges at the Si/SiO2 interface and introduce free carriers, and also change the refractive index and light absorption through plasma dispersion effects [19]. Displacement damage can cause lattice defects in the target material, which will change the optical properties through local heat and pressure spikes produced by displacement defects [20,21]. Thus, there remains an on-going important consideration about radiation-hardened design.

Investigating the radiation effects of silicon photonic devices under γ-ray is important for space and nuclear reactor environments. The deposited dose in a radiation-resistant enclosure of a nuclear reactor is up to 50 Mrad(Si), and in the reactor core, the dose rate is about 105 to 1010 rad/h [22]. Doses due to high-energy electrons and protons in space environments can range up to ∼1 Mrad(SiO2) in near-Earth space missions [1]; device sensitivities are often evaluated via gamma irradiation to equivalent total-ionizing-dose damage levels [23]. Low-energy protons are also commonly found in the space environment and can degrade devices more severely than high-energy protons [1,24].

The correlation between the radiation response of device and waveguide size has been demonstrated experimentally in our previous work, in which wider waveguides are found to be less sensitive to refractive index changes [13]. Therefore, we expect devices with 3 µm-thick silicon waveguide layers to be less influenced by irradiation. Silicon photonic devices, which are extensively used and fabricated on SOI wafers with top silicon thickness of 220 nm, are known to be susceptible to high energy irradiation [13]. Silicon photonic devices based on a 3 µm-thick SOI photonic integration platform is another important branch for silicon photonic integration. These devices have high coupling efficiency and ultralow propagation loss, as well as large fabrication tolerance in comparison with 220 nm top silicon waveguides [2527].

In this work, we design and fabricate MZIs and AWGs that are widely used in optical interconnection systems. And we evaluate the radiation response of Si passive photonic devices to high energy 60Co γ-ray irradiation up to 41 Mrad(Si) and 170-keV proton irradiation to a fluence up to 1×1016 proton/cm2. Spirals are utilized to evaluate the waveguide propagation loss change after irradiation. Compared with competitive technologies, the performances of 3 µm SOI devices are significantly improved.

2. Experiment and results

An SOI wafer with 3 µm top silicon and 0.4 µm buried silicon dioxide (BOX) is used as starting material in this work. Device structures are first patterned on the SOI wafer by deep-ultraviolet (DUV) photolithography. Then the devices are defined using dry etch. The etching recipes are particularly developed to reduce the roughness and improve the sharpness of the waveguide sidewall, which is critical for low loss waveguide. All devices in this work have ridge waveguides with height of 2.8 µm and width of 2 µm. 1.2 µm SiO2 and 0.7 µm silicon nitride layers are deposited by plasma-enhanced chemical vapor deposition as top cladding.

A cross-sectional view of the material stack (not to scale) is shown in Fig. 1(a). Here silicon nitride is deposited as a protective passivation layer. Figures 1(b) and (c) show the fundamental TE and TM mode profiles of the waveguide at a wavelength of 1550 nm performed by a Finite Difference Eigenmode (FDE) solver in Lumerical MODE solutions. Comparing the TE and TM mode shapes and effective refractive index (neff) values, the mode profiles are indistinguishable, and the neff values are nearly equal, indicating that the polarization state is nearly degenerate. The edge coupler is designed to filter out the energy of higher order modes. Therefore, the ridge waveguide in our work maintains a single-mode behavior and polarization independence for the key technology in silicon photonics integration. This platform can support deployment applications ranging from optical filters to data centers.

 figure: Fig. 1.

Fig. 1. (a) Cross-sectional view of material stack considered in this work. Mode profiles of (b) TE and (c) TM and effective refractive indices in ridge waveguide at 1550 nm used in this work performed by Finite Difference Eigenmode (FDE) solver in Lumerical MODE solutions.

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Figure 2(a) exhibits the cross-sectional view of a waveguide fabricated in the silicon photonic (SiPh) process platform, obtained via Scanning Electron Microscope (SEM), and showing the good etching profile, steepness, and wall roughness. Figures 2(b)-(d) show the images (top view) of fabricated devices obtained by an optical microscope. Figure 2(b) is the fabricated 1×4 AWG. The AWG is designed to use as a demultiplexer in a harsh radiation environment. And the AWG demultiplexer consists of an input waveguide (one channel), two free propagation regions (FPRs with diameter of 700 µm), and 35 arrayed waveguides with length difference of 1.6 µm between adjacent waveguides that are placed between two FPRs, and output waveguides. The output waveguides include four channels with channel spacing of 20 nm. The input light transports to the first FPR, and the diffracted light expands and couples into the arrayed waveguides with different phase differences due to constant optical length differences between adjacent waveguides. To obtain constructive interference in the output position, the accumulated optical path difference between the two adjacent arrayed waveguides is an integer multiple of the wavelength. Figure 2(c) shows the fabricated 1×2 MZI. The MZI is designed as an asymmetric structure with an imbalanced arm of 207 µm to provide a phase difference. A 1×2 multimode interferometer (MMI) is located as a beam splitter in the input port and a 2×2 MMI is employed as the beam combiner in the output port. An incoming light beam is split identically. Then two beams follow paths through imbalanced arms, with constructive or destructive interference occurring when the accumulated phase difference (Δφ) in the output port satisfies the following conditions, respectively:

$$\Delta \varphi \textrm{ = }\left\{ {\begin{array}{{l}} {2\pi m,\,\,m\textrm{ }is\textrm{ }an\textrm{ }integer}\\ {\pi n,\,\,\,n\textrm{ }is\textrm{ }an\textrm{ }odd\textrm{ }integer} \end{array}} \right.$$

 figure: Fig. 2.

Fig. 2. Devices fabricated by SiPh process platform. (a) SEM image of cross-sectional of waveguide. Optical images (top view) of (b) AWG, (c) MZI, (d) Spiral.

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Figure 2(d) shows three spirals with different lengths of 11.4 cm (SPI3), 7.06 cm (SPI2), and 3.2 cm (SPI1), respectively. All spirals are ridge waveguides with a bending radius of 250 µm; thus, the bending loss can be ignored. The Euler bending radius changes from 1500 µm to 150 µm gradually to reduce losses and avoid mode leakage.

This work focuses on the cumulative effects induced by irradiation; transient effects of irradiation are not considered in this study. All chips are completely and uniformly irradiated at room temperature. Four test chips are irradiated with 170-keV protons to fluences from 1×1013 to 1×1016 p/cm2. Four other chips are irradiated with 60Co γ-rays with average energy of 1.24 MeV to doses of 15 to 41 Mrad(Si), at a dose rate of 300 rad(Si)/s. Two chips are maintained as un-irradiated control samples. Figure 3 shows an overview of the optical measurement setup in this work. The pre- and post-irradiation devices are accurately characterized on a Fiber-Pro Auto Align workstation; a polarization synthesizer is employed to control the polarization state. Transmission spectra around 1310 nm and 1550 nm are scanned over the power meter and observed on the computer screen. An optical switch enables changes between different input tunable laser sources. The devices under test (DUTs) are kept at a constant temperature of (21 ± 0.2) °C by a thermostat stage. Edge coupling is employed by means of lensed fibers with a spot diameter of (2.5 ± 0.25) µm. Optical measurements are not performed in real time during exposure; samples are tested at the end of the irradiation sequences. Thus, no irradiation-induced changes occur in testing setup.

 figure: Fig. 3.

Fig. 3. Overview of the Auto-Align optical measurement setup in this work.

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The transmission spectra of 1×4 AWGs before (solid curve) and after (dash dot curve) 170-keV proton irradiation to fluences of 1×1013 p/cm2 and 1×1016 p/cm2 are presented in Figs. 4(a) and (b), respectively. The average measured insertion loss is about -1.7 dB, which is very close to the designed value (-1.5 dB). The crosstalk for channel wavelength at 1310 nm from the adjacent channel is about -30 dB. Insertion loss and crosstalk hardly change after proton irradiation. In addition, no significant changes in the linewidth and wavelength are observed; all deviations are within test limits. Similarly, 60Co γ-ray irradiation of AWGs also shows unchanged linewidth, insertion loss, and wavelength. Responses of these devices are improved over those of Silica-on-Silicon (SoS) AWGs exposed to 300 krad 60Co γ-ray irradiation [14], indicating that these devices are excellent candidates for application in harsh radiation environments.

 figure: Fig. 4.

Fig. 4. Transmission spectra of AWGs at O band before and after 170-keV proton irradiation to fluences of (a) 1×1013 p/cm2, (b) 1×1016 p/cm2.

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Figures 5(a) and (b) show transmission spectra of MZIs before (blue dashed line) and after (red solid line) 1×1016 p/cm2 proton and 41 Mrad (Si) 60Co γ-ray irradiation, respectively. Figures 5(c)-(e) exhibit detailed views at different wavelength ranges. After high energy irradiation, wavelengths experience a tiny shift (∼100 pm). Low insertion loss of -1 dB is demonstrated, and no discernible insertion loss change after radiation is found. A pronounced reduction of extinction ratio (ER) below 1550 nm after proton irradiation is observed in Figs. 5(a) and (c). The bonding of Si and H from proton implantation may be responsible for the change. Vibrational overtones of Si–H bonds associated with a conspicuous absorption peak near 1510 nm can be responsible for the reduction of ER [7,28].

 figure: Fig. 5.

Fig. 5. Transmission spectra of MZIs before (blue dashed line) and after (red solid line) (a) 170-keV proton irradiation to a fluence of 1×1016 p/cm2. (b) γ-ray irradiation with total dose of 41 Mrad(Si). (c-e) Detailed views at different wavelength ranges of the spectra.

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The tiny wavelength shift can be used to deduce variation of the effective refractive index of the waveguide mode via Eq. (2) [8]:

$$\Delta {n_{eff}} = \frac{{\Delta \lambda \cdot \lambda }}{{FSR \cdot \Delta L}}$$
where λ is the wavelength, Δλ is the measured wavelength shift, Δneff is the effective refractive index change of the waveguide mode. ΔL is the length of the imbalanced arm of MZI, 207 µm. From Eq. (2), Δneff is about 10−4, which is one order of magnitude smaller than those observed for silicon oxynitride MZIs exposed to 2.3 MeV alpha-irradiation to a fluence of 1015 cm-2 [7], and for 220 nm SOI MZIs under 1012 cm-2 neutron irradiation [13]. AWGs and MZIs experience the same mode refractive index variation of 10−4 due to the same waveguide size. The channel center wavelengths of AWGs are not influenced by irradiation, indicating that AWGs are more tolerant to the effective refractive index change induced by irradiation than MZIs.

Figure 6, with 95% confidence interval (CI) for the measured MZIs data, plots the calculated variation of effective refractive index of MZIs from Eq. (2) for proton irradiation to fluences of 1×1013 to 1×1016 p/cm2 (Fig. 6(a)), and 60Co γ-irradiation to doses of 15 to 41 Mrad(Si) (Fig. 6(b)), respectively. One of the key results that emerges from these findings is that Δneff decreases slightly with increasing fluence/dose in each case. Table 1 compares the results of this work with those based on different material systems. As shown, 3 µm SOI MZIs and AWGs in this work show more radiation-resistance than devices in 220 nm SOI [13], SiON-on-SiO2 [7], and SoS material systems [14].

 figure: Fig. 6.

Fig. 6. Variations of effective refractive index after (a) proton irradiation with total fluences from 1013 to 1016 p/cm2, (b) γ-ray irradiation with doses from 15 to 41 Mrad(Si). Insets are transmission spectra of MZIs before (blue curve) and after (red curve) irradiation.

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Tables Icon

Table 1. Comparison of MZIs and AWGs in this work with others based on different material systems

3. Discussion

The radiation responses of MZIs and AWGs from our experiment provide important insights into the radiation damage in 3 µm top silicon devices for photonic integrated applications. Both proton and gamma irradiation can produce defects in materials, which lead to optical property modifications. In passive silicon photonic waveguides, TID-induced charging of traps at the Si/SiO2 interface affects carrier concentrations in the crystalline silicon core layer. In addition, hydrogen can be released and react with pre-existing defect centers within the materials. Each of these effects can lead to degradation of optical properties [11,29]. Displacement damage can also cause irreversible damage in the waveguide. For crystalline silicon, the lattice defects can introduce energy levels and volumetric modification. For the amorphous cladding, radiation-induced compaction effects have been observed [7,8]. Both types of damage effects are taken into account in the irradiations of the 3 µm SOI optical devices.

Radiation-induced defects may increase propagation losses within the waveguide. From the above test results, slight changes in effective refractive index are observed. In contrast, no measurable changes are observed in radiation induced-optical transmission losses for both AWGs and MZIs after irradiation. As a further check for such losses, radiation responses on different spirals are shown in Fig. 7. Three spirals with different lengths are irradiated with 60Co γ-ray to 15 Mrad(Si) and 170-keV protons to a fluence of 1015 p/cm2. Figures 7(a) and (b) show the transmission losses of spirals before and after 60Co γ-ray irradiation, respectively. Figures 7(d) and (e) exhibit the transmission spectra of spirals before and after proton irradiation. The waveguide propagation losses measured by cutback method before and after irradiation are presented in Figs. 7(c) and (f), respectively. The initial propagation losses display differences originating from different chips, and show slight increases by about 0.07 dB/cm at O-band and 0.08 dB/cm at C-band after 15 Mrad(Si) 60Co γ-ray irradiation. Similarly, after 1015 p/cm2 proton irradiation, propagation losses increase by about 0.02 dB/cm at O-band and 0.03 dB/cm at C-band, indicating that the radiation-induced change in propagation loss is wavelength independent. Although a slight increase in propagation loss after irradiation is induced, the slightly increased propagation loss is negligible for AWG and MZI of length less than 1 cm in this work (within test limits).

 figure: Fig. 7.

Fig. 7. Transmission losses of spirals at (a) O band and (b) C band for 60Co γ-irradiation to 15 Mrad(Si), (c) Propagation losses before and after 60Co γ-irradiation to 15 Mrad(Si) measured by cutback method; transmission losses of spirals at (d) O band and (e) C band for proton irradiation for 1015 p/cm2, (f) Propagation losses before and after proton irradiation for 1015 p/cm2 measured by cutback method.

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Heavy ions such as protons can be shielded using aluminum [9]. However, gamma photons are hardly able to be shielded owing to strong penetrating ability. Analyzing the influence of gamma photons on the devices is imperative. Gamma rays are widely used to mimic ionizing damage with production of electron-hole pairs. The structural modification introduced by displacement damage is also an important factor to evaluate optical devices. Theoretically, displacement damage can also be introduced by energetic secondary electrons that are emitted via Compton scattering process. The energetic electrons can collide with the atoms of the target material, and then shift atoms from their lattice sites. In order to verify whether Compton scattering electrons can cause displacement damage in the devices considered in this work, we compute the maximum recoil energy of atoms colliding with Compton scattered electrons via the methods of Doyle et al. [30] and Du et al. [15]. The electrons are assumed to receive nearly all the energy of the gamma photons, and the average gamma photon energy is 1.24 MeV. Here, the relativistic equation is used to infer the maximum atomic recoil energy, since the energy of the electron is considerably larger than the rest energy of 0.511 MeV [15,30]:

$$\begin{array}{l} {T_{\max }}\textrm{ = }\frac{{2{{({\gamma {m_0}{\nu_0}} )}^2}}}{M} = 2\frac{{{m_0}}}{M}{\gamma ^2}{\beta ^2}{m_0}{c^2}\\ \gamma \textrm{ = }\frac{1}{{\sqrt {1\textrm{ - }{\beta ^2}} }} ,\kern 7pt \beta \textrm{ = }\frac{{{\nu _0}}}{c} \end{array}$$

Here ${T_{max}}$ is the maximum recoil energy of an atom, M represents the atom’s mass, m0 is electron rest mass, here is 0.000545 amu; $v$0 and c denote electron and light velocity, respectively. c is 3×108 m/s. v0 is calculated by the mass-energy equation, E = mc2. β is defined by v0 and c, γ is related with β. Parameters involved in Eq. (3) are given in Table 2. (1amu = 1.66×10−27 kg).

Tables Icon

Table 2. Parameters involved in Eq. (3)

Based on the Eq. (3), ${T_{max}}$ for Si, O, and N are 204 eV, 407 eV, and 357 eV, respectively. The displacement threshold energies for Si in the bulk silicon is about 20 eV [31], and are 33.5 eV for Si atom and 16.3 eV for O atom in a-SiO2 [32]. In silicon nitride, displacement threshold energies for Si and N atoms are 11 eV and 21 eV, respectively [33,34]. The calculated maximum recoil energies are significantly larger than the displacement threshold energies. Therefore, the probability of producing displacement defects by Compton scattering energetic electrons is considered to be relatively high.

Measured slight variation of effective refractive index of mode in MZIs can be explained by the combined effects of top cladding and silicon core, accounting for radiation-induced tiny shift of wavelength in MZIs spectrum. Here, the thin silicon nitride layer is not considered to be analyzed due to low confinement factors. For SiO2 upper cladding, this is caused primarily by compaction of the amorphous network after high energy irradiation [35]. Changes within the silicon core are caused primarily by the structural damage, originating from the tiny volumetric expansion after ion implantation into crystalline silicon, leading to a decrease in atomic density and corresponding changes in refractive index [36,37], as indicated by the circle in Fig. 6.

For microelectronic devices in space environments, lower-dose-rate irradiation can lead to significant changes in electrical response due to the complex interactions of trapped charge, defects in dielectric layers, and hydrogen transport and reactions [38,39]. Although dose-rate effects are generally less significant in optical materials than in microelectronics, future work in which a broader range of dose rates than the rate of 300 rad(Si)/s chosen here would be useful to provide more information on how closely the results of laboratory tests can predict the response in lower-rate space environments. Characterization of single-event effects (SEEs) (e.g., cosmic-ray or proton-induced transients) is also important for the evaluation of optical devices for space applications [40,41].

4. Conclusions

In summary, we present radiation-resistant silicon photonic passive devices based on 3 µm-thick silicon-on-insulator photonic integration platform, including low-loss arrayed waveguide gratings and Mach-Zehnder interferometers that are extensively used in photonic devices and integrated chips. The designed AWGs and MZIs show good performances of low insertion loss, high ER, and low crosstalk for AWGs. We have investigated their radiation responses under proton and 60Co γ-ray irradiation. For devices with length of less than 1 cm, the propagation loss post-irradiation can be negligible. AWGs are confirmed to have stable photonic performance after a total fluence as high as 1×1016 p/cm2 proton irradiation and 60Co gamma photons with 41 Mrad(Si) absorbed dose. For MZIs, linewidth and losses hardly change, with negligible minimal wavelength shifts. Variation in effective refractive index of devices under irradiation are attributed most likely to radiation-induced compaction of amorphous network and structural modification, which volumetric changes induced by proton and Compton scattered electrons. Such a 3 µm-thick silicon-on-insulator waveguide is beneficial in improving the radiation tolerance of photonic components in harsh radiation environments.

Funding

Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021232); National Natural Science Foundation of China (61905269); Shanghai Sailing Program (19YF1456600).

Acknowledgments

The authors thank the support by Youth Innovation Promotion Association CAS, National Natural Science Foundation of China and Shanghai Sailing Program.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)

Fig. 1.
Fig. 1. (a) Cross-sectional view of material stack considered in this work. Mode profiles of (b) TE and (c) TM and effective refractive indices in ridge waveguide at 1550 nm used in this work performed by Finite Difference Eigenmode (FDE) solver in Lumerical MODE solutions.
Fig. 2.
Fig. 2. Devices fabricated by SiPh process platform. (a) SEM image of cross-sectional of waveguide. Optical images (top view) of (b) AWG, (c) MZI, (d) Spiral.
Fig. 3.
Fig. 3. Overview of the Auto-Align optical measurement setup in this work.
Fig. 4.
Fig. 4. Transmission spectra of AWGs at O band before and after 170-keV proton irradiation to fluences of (a) 1×1013 p/cm2, (b) 1×1016 p/cm2.
Fig. 5.
Fig. 5. Transmission spectra of MZIs before (blue dashed line) and after (red solid line) (a) 170-keV proton irradiation to a fluence of 1×1016 p/cm2. (b) γ-ray irradiation with total dose of 41 Mrad(Si). (c-e) Detailed views at different wavelength ranges of the spectra.
Fig. 6.
Fig. 6. Variations of effective refractive index after (a) proton irradiation with total fluences from 1013 to 1016 p/cm2, (b) γ-ray irradiation with doses from 15 to 41 Mrad(Si). Insets are transmission spectra of MZIs before (blue curve) and after (red curve) irradiation.
Fig. 7.
Fig. 7. Transmission losses of spirals at (a) O band and (b) C band for 60Co γ-irradiation to 15 Mrad(Si), (c) Propagation losses before and after 60Co γ-irradiation to 15 Mrad(Si) measured by cutback method; transmission losses of spirals at (d) O band and (e) C band for proton irradiation for 1015 p/cm2, (f) Propagation losses before and after proton irradiation for 1015 p/cm2 measured by cutback method.

Tables (2)

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Table 1. Comparison of MZIs and AWGs in this work with others based on different material systems

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Table 2. Parameters involved in Eq. (3)

Equations (3)

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Δ φ  =  { 2 π m , m   i s   a n   i n t e g e r π n , n   i s   a n   o d d   i n t e g e r
Δ n e f f = Δ λ λ F S R Δ L
T max  =  2 ( γ m 0 ν 0 ) 2 M = 2 m 0 M γ 2 β 2 m 0 c 2 γ  =  1 1  -  β 2 , β  =  ν 0 c
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