rus
Issue #3/2022
D. A. Koroteev, V. S. Gerasimenko, N. D. Gerasimenko, V. M. Petrov
The Role of Leaky Modes in the Operation of Devices Based on Integrated-Optical Mach-Zehnder Interferometers
The Role of Leaky Modes in the Operation of Devices Based on Integrated-Optical Mach-Zehnder Interferometers
DOI: 10.22184/1993-7296.FRos.2022.16.3.236.244
The effect of leaky modes on the operation of RF integrated-optical amplitude modulators and quantum noise generators made according to the Mach-Zehnder interferometer scheme has been studied. It is shown that the optical power dissipation through the leaky modes leads to a decrease in the dynamic range of the amplitude modulator, and, most importantly, to the possible intruder’s attack on a quantum communication line. In relation to the quantum noise generators made according to the Mach-Zehnder interferometer scheme, the leaky modes ensure penetration of vacuum fluctuations into the noise generator circuit that is necessary for the generator operation.
The effect of leaky modes on the operation of RF integrated-optical amplitude modulators and quantum noise generators made according to the Mach-Zehnder interferometer scheme has been studied. It is shown that the optical power dissipation through the leaky modes leads to a decrease in the dynamic range of the amplitude modulator, and, most importantly, to the possible intruder’s attack on a quantum communication line. In relation to the quantum noise generators made according to the Mach-Zehnder interferometer scheme, the leaky modes ensure penetration of vacuum fluctuations into the noise generator circuit that is necessary for the generator operation.
Теги: attacks on the quantum communication lines integrated-optical amplitude modulators quantum noise generators tunnelling modes атаки на квантовые линии связи интегрально-оптические амплитудные модуляторы квантовые генераторы шума моды утечки
The Role of Leaky Modes in the Operation of Devices Based on Integrated-Optical Mach-Zehnder Interferometers
D. A. Koroteev, V. S. Gerasimenko, N. D. Gerasimenko, V. M. Petrov
ITMO National Research University Saint-Petersburg, Russia
The effect of leaky modes on the operation of RF integrated-optical amplitude modulators and quantum noise generators made according to the Mach-Zehnder interferometer scheme has been studied. It is shown that the optical power dissipation through the leaky modes leads to a decrease in the dynamic range of the amplitude modulator, and, most importantly, to the possible intruder’s attack on a quantum communication line. In relation to the quantum noise generators made according to the Mach-Zehnder interferometer scheme, the leaky modes ensure penetration of vacuum fluctuations into the noise generator circuit that is necessary for the generator operation.
Keywords: integrated-optical amplitude modulators, quantum noise generators, tunnelling modes, attacks on the quantum communication lines
Received on: 06.04.2022
Accepted on: 20.04.2022
1. Introduction, problem statement
The integrated-optical interferometers made according to the Mach-Zehnder interferometer (MZI) scheme are used to implement the super-broadband quantum noise generators [1, 2], high-speed amplitude modulators [3], quantum photonic processors [4], and other high-precision and broadband devices based on the integrated optics [5]. At present, particular attention is drawn to such devices due to the rapid development of integrated optical chips for producing various controlled quantum photon gates [6] and computers [7, 8]. The literature data analysis shows that all these devices are based on one or several interconnected MZIs that have one or two inputs and one or two outputs. The most typical cases are shown in Fig. 1.
The occurrence of leaky modes shown in Fig. 1b, c is a consequence of the need to obtain the unitarity of energy transfer from the device input to the output, and in a more general case, the need to implement the energy conservation law. Indeed, when the phase difference in the interferometer arms is equal to π, the interferometer output power Out1 is equal to zero. In this case, the optical power is not supplied to the output, but it is dissipated by the tunnelling modes.
It appears that for the first time, the leaky modes were described in the works of Markowitz [9] and Berown [10, 11]. In his paper, Markowitz proposes to use the leaky modes as a solution to the wave equation. The leaky modes were discovered experimentally by Hall and Ye [12]. In this experiment, a three-layer ZnSe waveguide on a GaSe substrate was used where only tunnelling modes can theoretically exist. During the experiment, the photographs of TE and TM modes were obtained for the substrates with various thicknesses. Further, the behavior of leaky modes in the waveguides of various shapes was described by Snyder and Love [13]. They also demonstrated the solutions to the field propagation problems in a waveguide using the tunnelling modes. Thus, the leaky modes are a well-studied and theoretically described phenomenon with an experimental confirmation.
When performing numerous works for assembly and adjustment of various integrated-optical interferometers, we were also able to repeatedly and directly observe the “output” of optical power from the MZI in the directions determined by the leaky modes.
In this paper, we provide the simulation results of leaky modes in the electrically controlled integrated-optical MZIs made on the surface of a lithium niobate (LiNbO3) crystal substrate.
2. Simulation results
We have studied the following devices, the architecture of which is similar to the MZI architecture. The common property is that all these devices have a noticeably prolate shape: the length-to-width ratio of the substrate can be 30 times or more. The substrate material is lithium niobate (LiNbO3), the waveguides are made using the titanium ion diffusion method.
2.1. MZI-based amplitude modulator
The MZI-based integrated-optical amplitude modulator can be considered as two back-to-back Y-couplers (see Fig. 2). The substrate length L can be 50–60 mm, the substrate width h can be 3–6 mm. The angle α is about 3°. The circuit with one input and one output is considered below.
When the voltage is applied to a pair of electrodes (2), it is possible to change the phase difference value Δϕ between the two interferometer arms in the range 0–π. The optical power Iin is supplied to the device input at the operating wavelength. If Δϕ = 0, then the output power Iout = Iin. If Δϕ = π, then the output power Iout = 0. In this case, the power is dissipated into the substrate. This dissipation occurs using the tunnelling modes.
We have performed a numerical calculation and analysis of the leaky modes for the device considered in this paper. Figure 3 shows a typical example of the amplitude distribution of the tunnelling mode for the output Y-coupler for the Δϕ = π case. It can be seen that in this case, the power propagating along the interferometer arms does not enter the output waveguide, but is completely fed to the substrate along two symmetric tunnelling modes. It further falls on the back face of the substrate. The angle between the waveguide axis and the tunnelling mode direction along which the maximum power is propagating is equal to 1°‑2°. Having considered the refractive index value of lithium niobate n ≈ 2,17 at a wavelength from the range of 1520–1560 nm, it is possible to assess the value of the Fresnel amplitude reflection coefficient r, determined by the difference in the refractive indices at an incidence angle close to normal one:
, (1)
where n1 is the refractive index of lithium niobate, n2 is the refractive index of air. Then the power reflected back into the substrate can be estimated at about 14% of the modulator input power.
Figure 4 shows an example of the propagation process simulation for the backward-reflected power from the back face of the substrate, with due regard to the angle α = 2.7°. This distribution is performed in zigzag fashion with the alternating peaks and bases.
The final optical power distribution at the input face of the modulator depends on the exact length-to-width ratio of a particular device and therefore cannot be determined with high accuracy. More important for us is the fact that an uncontrolled optical power propagation occurs in the modulator substrate as a result of phase modulation in one of the MZI arms that can reach 14% of the input power. Obviously, part of this power can again enter the waveguides, for example, through the leaky modes of the input Y-coupler. This leads to a noise level increase at the MZI output, and, ultimately, to a decrease in the dynamic range that is observed in the field.
2.2. MZI-based quantum noise generator
The operating principle of such a device is described by us in [1, 2]. In this paper we consider how the vacuum fluctuations penetrate the quantum noise generator circuit. Two flow charts of the quantum noise generator are shown in Fig. 5. Fig. 5a shows a symmetrical circuit (two inputs and two outputs). In such a circuit, the condition for the transformation unitarity is met, there are no leaky modes, and the vacuum fluctuations enter the circuit through the second open input.
The second flow chart (Fig. 5b) is asymmetrical (one input and two outputs). The leaky modes occur due to the need to comply with the transformation unitarity condition. The vacuum fluctuations enter it through the leaky modes of the input Y-coupler.
Figure 6 shows an example of numerical simulation for the phase distribution of the vacuum fluctuation field Evak through the leaky modes at the asymmetrical circuit input. The local oscillator field ELO enters the noise generator through the input waveguide. It should be noted that it is the possibility of vacuum fluctuations penetrating into the waveguide through the substrate that ensures operation of the quantum noise generators made according to this chart.
3. Results and discussion
The dynamic range of the amplitude modulator is determined “from below” by the noise level, and “from above” by the signal strength at which the higher harmonics occur [5]. In this paper, we consider the noise impact. The paper [3] provides the experimental data. It follows that the experimentally measured dynamic range is no more than 70–80 dB that is much less than the theoretical estimates, according to which the theoretical threshold is about 130 dB [5]. In this case, it should be considered that in fact, the dynamic range of the entire laser-modulator-photodetector system is measured in practice. Each of these elements contributes to the noise level. In addition, an important factor is coordination of all path elements in terms of the inherent noise level. Therefore, the noise resulting from the leaky modes is only one of noises contributing to the loss of dynamic range. However, they must be considered and, if possible, minimized.
The leaky modes are of parasitic nature for the MZI-based amplitude modulators. They create practically uncontrolled optical noise in the substrate that ultimately reduces the dynamic range of the modulator. In this case, it is possible to propose application of an antireflection coating to the end faces of the substrate as a preventive measure that should reduce the backward reflection of the optical power. The back side of the substrate should be covered by an absorbing coating to reduce the power propagation in the substrate.
Another problem that is no less serious and related to the availability of leaky modes is the possible attack on a quantum communication line that uses both phase and amplitude modulators (Fig. 6). The optical power dissipated through the tunnelling modes is absolutely synchronized with the AM strokes used on the line. It can be assumed that some part of the dissipated power can be detected by an intruder in a quantum communication line at a noticeable distance from the modulator. Having considered that the power dissipation can reach 14% (≈ –8.54 dB) of the input power, this scenario seems quite realistic. In this case, in addition to the problem of reducing the optical power coming out through the tunnelling modes at an amplitude modulation, there is a problem of using the coding, or modulation type, that is the most energetically favorable. In the case of “normal” amplitude modulation, part of the transmitted power is dissipated into the substrate.
For a quantum noise generator made according to an asymmetric circuit, availability of leaky modes ensures that the field of vacuum fluctuations enters the circuit. In this case, the crucial task is to determine such an amplitude-phase mode distribution that would provide the maximum power of vacuum fluctuations.
Conclusion
The simulation results of leaky modes in the electrically controlled integrated-optical MZIs, made on the surface of the lithium niobate (LiNbO3) crystal substrate, are presented. It is shown that the leaky modes are of parasitic nature for the MZI-based amplitude modulators, and the optical power dissipation through the leaky modes leads to a decrease in the dynamic range of the amplitude modulator. The occurrence of leaky modes increases the likelihood of an attack on a quantum communication line that uses both phase and amplitude modulators.
ABOUT AUTHORS
Viktor Petrov, Doctor of Physical and Mathematical Sciences (Radiophysics), Doctor of Physical and Mathematical Sciences (Optics) vmpetrov@itmo.ru, Chief Researcher, National Research University ITMO, St. Petersburg, Russia.
ORCID: 0000-0002-8523-0336
Gerasimenko Natalya Dmitrievna, Engineer, National Research University ITMO, Faculty of Photonics and Optoinformatics, St. Petersburg, Russia.
ORCID: 0000-0002-6039-9485
Gerasimenko Vladislav Sergeevich, Engineer, Faculty of Photonics and Optoinformatics, National Research University ITMO, St. Petersburg, Russia.
ORCID: 0000-0002-9709-3850
Koroteev Denis Aleksandrovich, student, National Research University ITMO, St.-Petersburg, Russia.
ORCID: 0000-0002-5489-4017
Contribution by the members
of the team of authors
Koroteeva D. A. personal contribution: choice of software, calculations, modeling, analysis of the results.
Personal contribution of Gerasimenko N. S.: Assembly, tuning and testing of experimental models of modulators.
Personal contribution of Gerasimenko V. S.: detection and visual observation of leak modes.
Personal contribution of V. M. Petrov: setting the problem, analyzing the results, supervising the work.
Conflict of interest
The authors claim that they have no conflict of interest. All authors took part in writing the article and supplemented the manuscript in part of their work.
D. A. Koroteev, V. S. Gerasimenko, N. D. Gerasimenko, V. M. Petrov
ITMO National Research University Saint-Petersburg, Russia
The effect of leaky modes on the operation of RF integrated-optical amplitude modulators and quantum noise generators made according to the Mach-Zehnder interferometer scheme has been studied. It is shown that the optical power dissipation through the leaky modes leads to a decrease in the dynamic range of the amplitude modulator, and, most importantly, to the possible intruder’s attack on a quantum communication line. In relation to the quantum noise generators made according to the Mach-Zehnder interferometer scheme, the leaky modes ensure penetration of vacuum fluctuations into the noise generator circuit that is necessary for the generator operation.
Keywords: integrated-optical amplitude modulators, quantum noise generators, tunnelling modes, attacks on the quantum communication lines
Received on: 06.04.2022
Accepted on: 20.04.2022
1. Introduction, problem statement
The integrated-optical interferometers made according to the Mach-Zehnder interferometer (MZI) scheme are used to implement the super-broadband quantum noise generators [1, 2], high-speed amplitude modulators [3], quantum photonic processors [4], and other high-precision and broadband devices based on the integrated optics [5]. At present, particular attention is drawn to such devices due to the rapid development of integrated optical chips for producing various controlled quantum photon gates [6] and computers [7, 8]. The literature data analysis shows that all these devices are based on one or several interconnected MZIs that have one or two inputs and one or two outputs. The most typical cases are shown in Fig. 1.
The occurrence of leaky modes shown in Fig. 1b, c is a consequence of the need to obtain the unitarity of energy transfer from the device input to the output, and in a more general case, the need to implement the energy conservation law. Indeed, when the phase difference in the interferometer arms is equal to π, the interferometer output power Out1 is equal to zero. In this case, the optical power is not supplied to the output, but it is dissipated by the tunnelling modes.
It appears that for the first time, the leaky modes were described in the works of Markowitz [9] and Berown [10, 11]. In his paper, Markowitz proposes to use the leaky modes as a solution to the wave equation. The leaky modes were discovered experimentally by Hall and Ye [12]. In this experiment, a three-layer ZnSe waveguide on a GaSe substrate was used where only tunnelling modes can theoretically exist. During the experiment, the photographs of TE and TM modes were obtained for the substrates with various thicknesses. Further, the behavior of leaky modes in the waveguides of various shapes was described by Snyder and Love [13]. They also demonstrated the solutions to the field propagation problems in a waveguide using the tunnelling modes. Thus, the leaky modes are a well-studied and theoretically described phenomenon with an experimental confirmation.
When performing numerous works for assembly and adjustment of various integrated-optical interferometers, we were also able to repeatedly and directly observe the “output” of optical power from the MZI in the directions determined by the leaky modes.
In this paper, we provide the simulation results of leaky modes in the electrically controlled integrated-optical MZIs made on the surface of a lithium niobate (LiNbO3) crystal substrate.
2. Simulation results
We have studied the following devices, the architecture of which is similar to the MZI architecture. The common property is that all these devices have a noticeably prolate shape: the length-to-width ratio of the substrate can be 30 times or more. The substrate material is lithium niobate (LiNbO3), the waveguides are made using the titanium ion diffusion method.
2.1. MZI-based amplitude modulator
The MZI-based integrated-optical amplitude modulator can be considered as two back-to-back Y-couplers (see Fig. 2). The substrate length L can be 50–60 mm, the substrate width h can be 3–6 mm. The angle α is about 3°. The circuit with one input and one output is considered below.
When the voltage is applied to a pair of electrodes (2), it is possible to change the phase difference value Δϕ between the two interferometer arms in the range 0–π. The optical power Iin is supplied to the device input at the operating wavelength. If Δϕ = 0, then the output power Iout = Iin. If Δϕ = π, then the output power Iout = 0. In this case, the power is dissipated into the substrate. This dissipation occurs using the tunnelling modes.
We have performed a numerical calculation and analysis of the leaky modes for the device considered in this paper. Figure 3 shows a typical example of the amplitude distribution of the tunnelling mode for the output Y-coupler for the Δϕ = π case. It can be seen that in this case, the power propagating along the interferometer arms does not enter the output waveguide, but is completely fed to the substrate along two symmetric tunnelling modes. It further falls on the back face of the substrate. The angle between the waveguide axis and the tunnelling mode direction along which the maximum power is propagating is equal to 1°‑2°. Having considered the refractive index value of lithium niobate n ≈ 2,17 at a wavelength from the range of 1520–1560 nm, it is possible to assess the value of the Fresnel amplitude reflection coefficient r, determined by the difference in the refractive indices at an incidence angle close to normal one:
, (1)
where n1 is the refractive index of lithium niobate, n2 is the refractive index of air. Then the power reflected back into the substrate can be estimated at about 14% of the modulator input power.
Figure 4 shows an example of the propagation process simulation for the backward-reflected power from the back face of the substrate, with due regard to the angle α = 2.7°. This distribution is performed in zigzag fashion with the alternating peaks and bases.
The final optical power distribution at the input face of the modulator depends on the exact length-to-width ratio of a particular device and therefore cannot be determined with high accuracy. More important for us is the fact that an uncontrolled optical power propagation occurs in the modulator substrate as a result of phase modulation in one of the MZI arms that can reach 14% of the input power. Obviously, part of this power can again enter the waveguides, for example, through the leaky modes of the input Y-coupler. This leads to a noise level increase at the MZI output, and, ultimately, to a decrease in the dynamic range that is observed in the field.
2.2. MZI-based quantum noise generator
The operating principle of such a device is described by us in [1, 2]. In this paper we consider how the vacuum fluctuations penetrate the quantum noise generator circuit. Two flow charts of the quantum noise generator are shown in Fig. 5. Fig. 5a shows a symmetrical circuit (two inputs and two outputs). In such a circuit, the condition for the transformation unitarity is met, there are no leaky modes, and the vacuum fluctuations enter the circuit through the second open input.
The second flow chart (Fig. 5b) is asymmetrical (one input and two outputs). The leaky modes occur due to the need to comply with the transformation unitarity condition. The vacuum fluctuations enter it through the leaky modes of the input Y-coupler.
Figure 6 shows an example of numerical simulation for the phase distribution of the vacuum fluctuation field Evak through the leaky modes at the asymmetrical circuit input. The local oscillator field ELO enters the noise generator through the input waveguide. It should be noted that it is the possibility of vacuum fluctuations penetrating into the waveguide through the substrate that ensures operation of the quantum noise generators made according to this chart.
3. Results and discussion
The dynamic range of the amplitude modulator is determined “from below” by the noise level, and “from above” by the signal strength at which the higher harmonics occur [5]. In this paper, we consider the noise impact. The paper [3] provides the experimental data. It follows that the experimentally measured dynamic range is no more than 70–80 dB that is much less than the theoretical estimates, according to which the theoretical threshold is about 130 dB [5]. In this case, it should be considered that in fact, the dynamic range of the entire laser-modulator-photodetector system is measured in practice. Each of these elements contributes to the noise level. In addition, an important factor is coordination of all path elements in terms of the inherent noise level. Therefore, the noise resulting from the leaky modes is only one of noises contributing to the loss of dynamic range. However, they must be considered and, if possible, minimized.
The leaky modes are of parasitic nature for the MZI-based amplitude modulators. They create practically uncontrolled optical noise in the substrate that ultimately reduces the dynamic range of the modulator. In this case, it is possible to propose application of an antireflection coating to the end faces of the substrate as a preventive measure that should reduce the backward reflection of the optical power. The back side of the substrate should be covered by an absorbing coating to reduce the power propagation in the substrate.
Another problem that is no less serious and related to the availability of leaky modes is the possible attack on a quantum communication line that uses both phase and amplitude modulators (Fig. 6). The optical power dissipated through the tunnelling modes is absolutely synchronized with the AM strokes used on the line. It can be assumed that some part of the dissipated power can be detected by an intruder in a quantum communication line at a noticeable distance from the modulator. Having considered that the power dissipation can reach 14% (≈ –8.54 dB) of the input power, this scenario seems quite realistic. In this case, in addition to the problem of reducing the optical power coming out through the tunnelling modes at an amplitude modulation, there is a problem of using the coding, or modulation type, that is the most energetically favorable. In the case of “normal” amplitude modulation, part of the transmitted power is dissipated into the substrate.
For a quantum noise generator made according to an asymmetric circuit, availability of leaky modes ensures that the field of vacuum fluctuations enters the circuit. In this case, the crucial task is to determine such an amplitude-phase mode distribution that would provide the maximum power of vacuum fluctuations.
Conclusion
The simulation results of leaky modes in the electrically controlled integrated-optical MZIs, made on the surface of the lithium niobate (LiNbO3) crystal substrate, are presented. It is shown that the leaky modes are of parasitic nature for the MZI-based amplitude modulators, and the optical power dissipation through the leaky modes leads to a decrease in the dynamic range of the amplitude modulator. The occurrence of leaky modes increases the likelihood of an attack on a quantum communication line that uses both phase and amplitude modulators.
ABOUT AUTHORS
Viktor Petrov, Doctor of Physical and Mathematical Sciences (Radiophysics), Doctor of Physical and Mathematical Sciences (Optics) vmpetrov@itmo.ru, Chief Researcher, National Research University ITMO, St. Petersburg, Russia.
ORCID: 0000-0002-8523-0336
Gerasimenko Natalya Dmitrievna, Engineer, National Research University ITMO, Faculty of Photonics and Optoinformatics, St. Petersburg, Russia.
ORCID: 0000-0002-6039-9485
Gerasimenko Vladislav Sergeevich, Engineer, Faculty of Photonics and Optoinformatics, National Research University ITMO, St. Petersburg, Russia.
ORCID: 0000-0002-9709-3850
Koroteev Denis Aleksandrovich, student, National Research University ITMO, St.-Petersburg, Russia.
ORCID: 0000-0002-5489-4017
Contribution by the members
of the team of authors
Koroteeva D. A. personal contribution: choice of software, calculations, modeling, analysis of the results.
Personal contribution of Gerasimenko N. S.: Assembly, tuning and testing of experimental models of modulators.
Personal contribution of Gerasimenko V. S.: detection and visual observation of leak modes.
Personal contribution of V. M. Petrov: setting the problem, analyzing the results, supervising the work.
Conflict of interest
The authors claim that they have no conflict of interest. All authors took part in writing the article and supplemented the manuscript in part of their work.
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