Generation of Optical Frequency Harmonics for Quantum Communication Systems at Side Frequencies
For the first time, a significant effect of passive (i. e., without additional amplification) optical feedback on the number of spectral components of the output signal of a phase microwave integrated optical modulator has been demonstrated. The conditions are revealed under which the number of additional spectral components increases to 2 × 20 pieces, and the spectral range covered by them reaches up to 69 GHz.
It is proposed to use this effect to increase the transmission rate of the quantum key, as well as to expand the functionality of optical communication systems using based on DWDM standards.
Accepted on: 19.11.2020
In a previous article , we described microwave integrated optical modulators that we developed for quantum communication systems . We have developed both types of high-quality modulators: phase (FM) and amplitude (AM). According to the algorithm, a pair of side frequencies that occur during phase modulation at frequency F of the optical carrier at frequency f is usually used to transmit a quantum key. The FM modulator is used for this purpose. The AM modulator provides input of the transmitted information, i. e. the code itself.
One of the main tasks from the point of view of applications is to increase the transmission speed of the quantum key. Obviously, it is possible to increase the transmission speed of the quantum key by increasing the number of generated side frequency pairs, N = 1, 2, 3 ….
In this paper, we will demonstrate a simple and effective method for increasing side frequency pairs without using an additional optical amplifier.
The idea of using feedback (both electrical and optical) in combination with a microwave optical modulator was proposed back in the 90s of the last century [3–6]. Radiation from a laser source was introduced into a ring waveguide resonator, in which it repeatedly passed through a modulator and an amplifier, after which a part of the radiation fell on an optical spectrum analyzer and a photodiode .
As seen in Fig. 1a, it was proposed to use an X-beam splitter (divider 2 by 2) as an input-output element in the resonator. The lower input and output of the beam splitter are used to create a feedback loop, the upper output is used to output radiation. After radiation is removed from the waveguide loop, it must be fed to another divider, from where it is directed to the spectrum analyzer and to the photodiode. The signal from the latter is fed to the electronic control unit of the modulator, where it is used to adjust the loop not only for microwave, but also for optical resonance. The task of the fiber amplifier is to compensate for losses in the loop, including the removal of some of the energy at the divider. Fig. 1b shows the calculated frequency combs for different resonator loop gains.
Further experiments using a similar scheme (Fig. 2a) made it possible to cover a wider spectral interval with a frequency comb of more uniform intensity (Fig. 2b) .
Another interesting way of creating feedback was converting the modulator into a Fabry-Perot cavity by placing it between Bragg mirrors that reflect part of the side spectral lines on one side of the optical carrier [5–7]. Some of these schemes also used in-resonator amplifiers, which allowed for the appearance of several additional spectral peaks using a single-port phase modulator .
The scheme of the experiment with Bragg mirrors is shown in Fig. 3a. The phase modulator is placed between the Bragg mirrors, as a result of which feedback occurs and side frequencies are generated (Fig. 3b).
It is important to note that the use of amplifiers inside the cavity leads to the fact that the optical noise at natural frequencies begins to amplify and harms the quality of the comb lines . That is why it would be more interesting to consider the results of operation of circuits without amplification [6, 7]. They also use Bragg mirrors. In this regard, the resulting frequency combs acquired a spectral dip with a width of several peaks .
The diagram of the experiment without using an amplifier is shown in Fig. 4a. Here, an electrical microwave signal is fed to two electrical inputs of the phase modulator through a circulator and a power divider. The modulator itself is placed between the Bragg mirrors. As a result, the forward and backward optical waves are modulated by electrical signals (bidirectional pumping).
Fig. 4b shows the output spectrum with bidirectional pumping with a dip due to the reflection band of the mirrors. To compensate for this dip, in some experiments, a fiber-optic erbium amplifier was added to the optical channel, which was already located outside the cavity .
Fig. 5a shows a diagram of an experiment with an amplifier. The phase modulator is placed between the Bragg filters, and the amplifiers are located in front of the feedback lines (in front of the polarization controller) and behind it (immediately after the second filter). Fig. 5b shows the output spectrum: the dashed line indicates the unmodulated signal, the solid line indicates the modulated signal.
It can also be noted that these studies used two-port phase modulators that effectively interact with light regardless of the direction of the beam path.
All the schemes discussed above were used to obtain optical frequency combs. However, this problem was and is being solved now in other ways, we will also consider them. One of the solutions is the use of an electro-optical oscillator : a system consisting of an electro-optical modulator, a beam splitter that removes part of the radiation through a light filter to a photodiode, an electronic amplification and filtering system, and a microwave divider that returns a part of the processed signal from the photodiode to the electronic input of the modulator . Even in the original idea with the selection of one frequency on the photodetector after the RF amplifier on the modulator, higher-order harmonics can naturally arise. When several side frequencies return to the electronic feedback, the system modulates each of them. This significantly increases the number of spectral peaks.
Another method for obtaining frequency combs is based on the possibility, with a certain selection of the control voltage, to use a Mach-Zehnder modulator to equalize the intensity of its output spectrum . This idea is based on the possibility of adjusting the length of the arms of the Mach-Zehnder interferometer to reduce, if necessary, the intensity of even or odd harmonics. However, using only the spectrum equalization procedure is not enough – it is still necessary to obtain a sufficiently large number of comb peaks.
To solve this problem, modulator cascades were used [11–14]. In the simplest version, the light first passed through an amplitude modulator and then a phase one . In more complex experiments, which demonstrated smoother and wider combs, a cascade of two modulators was supplemented by one more: an amplitude one at the input of the stage  or a phase one at the output . The first of these cascades showed a wider ridge, but less intensity for each peak. In this regard, a further increase in the number of modulators was studied only for phase devices: in experiments with a cascade of three phase modulators with one amplitude modulator at the output, more than 70 spectral lines were obtained within –10 dB relative to the input power  (Fig. 6). Fig. 6 shows the output spectrum when using a cascade of modulators, consisting of three phase and one amplitude: from top to bottom: 73, 65 and 63 lines, respectively.
Another interesting area of research is lithium niobate film modulators [15–17]. The use of thin strip waveguides obtained by electron-beam lithography makes it possible to localize the light field in a much smaller section of the waveguide than when using traditional diffusion technology. This makes it possible to reduce the distance between the traveling wave electrodes.
In addition, there are technologies for placing such a waveguide not under the electrodes, but between them. All this makes it possible to achieve huge modulating field strengths in the region of the waveguide, and hence effective modulation with a large number of higher-order harmonics  (Fig. 7). Fig. 7a shows a schematic of the modulator, Fig. 7b shows the measured (red) and theoretical (blue) output spectra.
In addition, the same technology makes it possible to obtain integrated optical ring resonators of extremely high Q-factor, which themselves can act as sources of frequency combs due to the Kerr effect [16, 18] (Fig. 8).
Fig. 8a shows a schematic diagram of the experiment: a notch filter made of a fiber Bragg grating is placed in a fiber ring resonator. Due to the resonator, when the fundamental frequency is attenuated, energy is transferred to the side frequencies (Fig. 8b). The frequency comb period is controlled by introducing a detuning between the pumping frequency of the resonator and the filter parameters: changing the pumping frequency without changing the center frequency of the filter leads to a shift in the spectral components of higher orders.
However, it is also possible to combine the above-described microwave modulator and an integral ring resonator, which gives, albeit a less wide, but smoother comb  (Fig. 9). Fig. 9a shows the schematic of the modulator (optical waveguides are shown in black, electrodes are shown in yellow), Fig. 9b shows the output spectrum – over 900 lines.
Our analysis of the literature has shown that schemes for the optical generation of the comb function, which contain a microwave optical modulator and an optical feedback loop containing an optical amplifier, have been studied quite well. The presence of an optical amplifier in such schemes leads to a noticeable negative effect of intrinsic noise on the formation of the profile of the comb function.
It is important to stress attention, that, from the point of view of quantum communication systems, the main drawback of the described schemes is the presence of an optical amplifier. According to the” Cloning Theorem “ , light quanta cannot be copied and, consequently, amplified. This means that the schemes discussed above are fundamentally inapplicable for quantum communication systems.
Thus, the study of a circuit containing a microwave optical modulator and a feedback loop that does not contain an amplifier is an urgent task.
We have investigated the optical spectra at the output of a phase microwave integrated optical modulator (PM) of our own production . The installation diagram is illustrated in Fig. 10. We used a laser with a wavelength of 1 550 nm, a spectral line width of <1 MHz and an output power of 2 mW. The optical feedback loop was made in the form of a fiber segment L ≈ 8.43 m long. A standard optical connector was located approximately in the middle of the loop, which made it possible to “turn on” and “turn off” the feedback. When the feedback was switched on, most of the optical power from the output of the PM was fed to its input, which led to a significant increase in the number of higher harmonics in the signal spectrum.
In our experiments, two types of Y-branches were used: with a division ratio of 1:9 and 1:3. The greatest feedback effect was found for the 1:9 splitter (i. e. approximately 90% of the power was directed into the feedback loop). Fig. 11a shows the signal spectra for the case of the “normal” mode of operation of the PM, Fig. 11b – signal spectra for a nonlinear mode of operation with an included optical feedback loop. experimental dependences were obtained for the phase modulation frequency F = 1.725 GHz.
Similar experiments were carried out for a Y-branch with a coupling ratio of 1:3. In this case, all characteristic dependences were also observed, however, the influence of the feedback was much smaller.
The influence of the phase modulation frequency on the number of higher spectral components (at a fixed length of the feedback loop) (Fig. 12a) and the influence of the input power of the phase modulation signal (Fig. 12b) were investigated. Fig. 12a shows the dependence of the number of spectral components with the included feedback on the modulation frequency at a power of 25 dBm. It can be seen from the graph that the maximum number of spectral components was observed at a modulation frequency of 1.725 GHz and was up to 40 pieces (i. e., approximately 20 on each side of the carrier). Changing the modulation frequency even by 5 MHz led to the fact that the number of peaks was halved. Moreover, a clearly pronounced symmetry of the dependence was observed to decrease and increase the modulation frequency in the vicinity of the peak. An increase in the number of maxima at the edges of the study area is associated with the following peaks, i. e., a periodic recurrence of the dependence is observed.
Discussion of the results
We have demonstrated the effectiveness of using optical feedback to increase the number of higher spectral components. Our proposed scheme is extremely simple: only a piece of optical fiber is used as feedback, which favorably distinguishes our scheme from all others considered in the literature. The circuit we have demonstrated does not use amplifiers, which makes it possible to reduce the noise level compared to experiments using amplifiers. In this case, the resulting comb does not have spectral gaps associated with the method of feedback formation.
As shown in Fig. 11a, the periodic recurrence of the dependence of the number of spectral components on the modulation frequency is easily explained by the following considerations. For resonance to occur, it is necessary that an integer number of microwave wavelengths fit into the feedback loop length: l = c / F, where c is the speed of light in vacuum, F is the phase modulation frequency. In our experiment, l ≈ 0.17379 m, and it is easy to calculate that then 49 microwave wavelengths l fit into the cavity loop length (8.52 m) with good accuracy.
Let’s discuss several examples of the functionality of the scheme we proposed here. First, the “laser + microwave optical modulator with a feedback loop” system can be considered as a radiation source of a set of N coherent oscillations. In our experiments, the spectral interval ΔF was 1.725 GHz, and the number N reached a total of 40 pieces. Consequently, the entire spectral range reached 40 × 1.725 GHz = 69 GHz. Obviously, by changing the feedback length as needed, one can select the spectral interval ΔF so that it corresponds to the standard (H)DWDM frequency grid, i. e. steps of 25, 50, or 100 GHz. In this case, one laser can be used to provide operation in several frequency channels at once.
Another interesting possibility is to “fit” a large number of spectral channels at once in the band of one standard spectral channel. Both of these options are in demand in communication systems using the CDMA (Code Division Multiple Access) principle, or another similar method.
Secondly, since the microwave integrated optical modulators presented here are intended for use in quantum communication systems at side frequencies, the presence of a large (up to 20 pieces) number of higher harmonics opens up the possibility of transmitting information at various pairs of side frequencies. This is an additional level of data transmission protection.
Thirdly, a significant advantage of our technology is the ability to significantly reduce the requirements, and, consequently, the cost, to the microwave generator itself. So, for example, the ability to work at the 10th harmonic allows you to use a generator that creates a phase modulation frequency of F = 1 GHz, which is equivalent to operating at the first harmonic of the generator with a frequency of F = 10 GHz.
And, finally, the most important advantage: the absence of an optical amplifier in our scheme allows us to use it in quantum communication systems.
Thus, the scheme considered here for amplifying nonlinearities in microwave integrated optical modulators can significantly expand their functionality and reduce the cost as applied to quantum communication systems.
Viktor Petrov, Doctor of Physical and Mathematical Sciences (Radiophysics), Doctor of Physical and Mathematical Sciences (Optics) firstname.lastname@example.org, 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
Shamrаi Alexander Valerievich, Doctor of Physical and Mathematical Sciences, e-mail: Achamrai@mail.ioffe.ru, Head. lab. of Quantum Electronics Physicotechnical Institute named after A. F. Ioffe, St. Petersburg, Russia.
ORCID: 0000 0003 0292 8673
Il’ichev Igor Vladimirovich, candidate of chemical sciences, senior researcher, lab. of Quantum Electronics Physicotechnical Institute named after A. F. Ioffe, St. Petersburg, Russia.
ORCID: 0000 0001 7803 0630
Agruzov Petr Mikhailovich, junior researcher, lab. of Quantum Electronics Physicotechnical Institute named after A. F. Ioffe, St. Petersburg, Russia.
ORCID: 0000 0002 1248 7069
Lebedev Vladimir Vladimirovich, junior researcher, lab. of Quantum Electronics Physicotechnical Institute named after A. F. Ioffe, St. Petersburg, Russia.
ORCID: 0000 0003 0292 8673
Contribution by the members
of the team of authors
The article was prepared on the basis of many years of work by all members of the team of authors.
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.