rus
Issue #7/2023
A. L. Sokolov, V. M. Petrov, V. Yu. Venediktov, D. D. Reshetnikov
Axially Symmetric Hermite-Gaussian Beams and the BB84 Protocol for the Space-Earth Quantum Cryptography Channel
Axially Symmetric Hermite-Gaussian Beams and the BB84 Protocol for the Space-Earth Quantum Cryptography Channel
DOI: 10.22184/1993-7296.FRos.2023.17.7.542.555
The paper provides a description of Hermite-Gaussian beams with an axially symmetric polarization structure generated by a vector superposition of Hermite-Gaussian modes with the indices 10 and 01. It is shown that the axial polarization symmetry makes such beams insensitive to the rotations relative to the optical axis that makes such a modification of the well-known BB84 protocol preferred for the quantum space cryptography systems. The possible creation and detection of such beams within the framework of a polarization protocol transmission using the devices with a radial polarizer is discussed.
The paper provides a description of Hermite-Gaussian beams with an axially symmetric polarization structure generated by a vector superposition of Hermite-Gaussian modes with the indices 10 and 01. It is shown that the axial polarization symmetry makes such beams insensitive to the rotations relative to the optical axis that makes such a modification of the well-known BB84 protocol preferred for the quantum space cryptography systems. The possible creation and detection of such beams within the framework of a polarization protocol transmission using the devices with a radial polarizer is discussed.
Теги: axially symmetric polarization structure polarization protocol quantum cryptography radial polarizer spiral rotator аксиально-симметричная поляризационная структура квантовая криптография поляризационный протокол радиальный поляризатор спиральный вращатель
Axially Symmetric Hermite-Gaussian Beams and the BB84 Protocol for the Space-Earth Quantum Cryptography Channel
A. L. Sokolov 1, V. M. Petrov 2, V. Yu. Venediktov 2, D. D. Reshetnikov 2
JSC “Research-and-production corporation “Precision system and Instruments” (RPC PSI), Moscow, Russia
Saint-Petersburg State University, Saint-Petersburg, Russia
Department of Laser ³Measurement and Navigation Systems, St. Petersburg Electrotechnical University “LETI”, St. Petersburg, Russia
The paper provides a description of Hermite-Gaussian beams with an axially symmetric polarization structure generated by a vector superposition of Hermite-Gaussian modes with the indices 10 and 01. It is shown that the axial polarization symmetry makes such beams insensitive to the rotations relative to the optical axis that makes such a modification of the well-known BB84 protocol preferred for the quantum space cryptography systems. The possible creation and detection of such beams within the framework of a polarization protocol transmission using the devices with a radial polarizer is discussed.
Keywords: quantum cryptography, polarization protocol, axially symmetric polarization structure, radial polarizer, spiral rotator
Article received:04.09.2023
Article accepted:13.10.2023
Introduction
The development of quantum cryptography systems [1] is driven by the need to protect information in the up-to-date communication networks. The use of various types of spatially structured beams can be one of the efficient solutions to the problem of optical data transmission in real turbulence conditions, both through the classical and quantum communication channels.
At present, a large number of theoretical and experimental works has been performed in this area, and a large number of articles have been published. For example, the influence of atmospheric channel turbulence on the propagation of Laguerre-Gaussian and Bessel-Gaussian vector beams and on the channel capacity has been studied [2, 3]; the transmission of higher modes of Laguerre-Gaussian beams in the real conditions of urban turbulence over a distance of 1.6 km has been demonstrated using a wavelength of 809 nm [4]; a key distribution rate of at least 120 Mbit/s has been experimentally demonstrated using the spectral, polarization and orbital angular momentum multiplexing at a wavelength of 1550.12 nm (193.4 THz) [5]. In the last example, the simulation of atmospheric turbulence has been performed using two two-dimensional phase light modulators, providing the turbulence corresponding to the value of the Rytov parameter . The quantum key distribution through a vortex optical fiber with the length of 60 m has been experimentally confirmed. In this case, a pair of entangled photons has been used, obtained as a result of the spontaneous parametric scattering at a wavelength of 405 nm [6]. The works are actively performed to study the propagation of Gaussian and vortex beams under the atmospheric turbulence conditions over the significant distances, up to 1 000 m [7].
A special place is held by the data transmission issues over distances of more than a thousand kilometers using the low-orbiting spacecrafts with the appropriate equipment [8–11]. If back in 2017, the paper [9] reported the achieved key transmission speed of “several kilohertz” to a satellite located at a distance of 1 200 km from the Earth, then in 2021 it was already reported about an “integrated fiber-optic and satellite network” with a total length 4 600 km and the secret key transmission to the satellite at a speed of 47.8 kBit/s [10]. It shall be noted that the possible application of beams with an axially symmetric polarization structure in the space systems, i. e., having the spatial polarization modulation in a plane orthogonal to the propagation direction, as the base beams that form a polarization cryptographic key is shown in the paper [11].
Almost all well-known papers devoted to the quantum key distribution (QKD) apply the BB84 protocol that uses two bases, each of which contains two photon states. In the first basis, the photon is linearly polarized in a vertical or horizontal way (0° or 90°); in the second basis, the photon is linearly polarized in a diagonal way (45° or 135°). In particular, this is due to the instability of the “phase” protocols during the propagation of light in a turbulent atmosphere.
At the same time, it should be noted that the application of the BB84 protocol using bases based on linear polarization of photons for the tasks of low-orbit spacecraft has its difficulties associated with the need to fix the position of the plane of polarization of light at each moment of time, both transmitting and receiving systems on earth and in space. The analysis shows that in the transmitting optical and laser systems, the polarization condition is changed significantly for different points of the hemisphere [12]. In the case of the polarization protocol, this means the dependence of two coordinate systems rotated by 45° on the relative orientation of the transmitting telescope and the spacecraft.
This dependence can be eliminated if the beams with an axially symmetric polarization structure are used [13–15].
The separate tasks are both obtaining the beams with a given axial polarization structure and their detection. Their generation methods can be divided into two main ones: the first group include the intracavity methods, when the first-order modes are generated instead of the main laser mode [16, 17], and the extracavity methods using the diffractive optical elements [18]. The paper [19] has shown that the second-order beams are generated when a linearly polarized beam is reflected from an angle reflector [20, 21]. In particular, a second-order optical vortex is generated in the presence of a special interference coating of the faces. The beam detection issue can be solved either by using a device that acts as a radial polarizer [15], or by using a device similar to the diffraction sorter, for example, [22].
The purpose of this paper is to propose an implementation of the well-known BB84 protocol for a space-based quantum data transmission system. A feature of such an implementation shall be its invariance in relation to the rotation around the z axis coinciding with the beam propagation direction.
1. Beams with an axially symmetric polarization structure
In the case of an axially symmetric polarization structure, regardless of the radial coordinate in the beam cross-sectional plane, each azimuth value corresponds to a certain orientation of the vector oscillation plane that is changed so that when returning to the original azimuth value, this plane makes an integer number of revolutions. The polarization-symmetric structures have two modifications depending on the rotation direction of the vector oscillation plane .
The polarization structure of these beams is invariant to the rotation relative to the beam axis: the polarization condition is maintained along the radius vector r for an arbitrary azimuth angle ϕ (Fig. 1).
In this paper, the beams with an axisymmetric polarization structure are proposed to be used when transmitting the quantum keys in outer space using the BB84 protocol.
Table 1 shows how the axially symmetric structures that make up the set for the modified polarization protocol are generated.
The basic orthogonal polarization structures of the polarization protocol are the beams formed by a vector superposition of linearly polarized Hermite-Gaussian modes with the indices 10 and 01 (Table 1). This is a radial polarization structure, since the vector at each point of the transverse plane is oriented along the radius (RP-beam), and an azimuth polarization structure, since the vector is directed at each point at a tangent to the concentric circles (AP-beam), and two orthogonal polarization structures with axial symmetry that are rotated at 45° relative to the RP-beam and AP-beam: right-twisted (RTP-beam) and left-twisted (LTP-beam). The Jones vectors in Table 1 are recorded in a cylindrical basis, where ϕ is the azimuth angle.
It is convenient to demonstrate the interaction of basic beams with the radial polarization elements in a special polarization (spiral) basis.
2. Spiral bases
For the convenience of mathematical manipulations, the spiral bases that are specified using two matrices [8] are used:
(1)
where ϕ is the azimuth angle, measured from the so-called neutral semi-axis, in the direction of which the matrices become unitary.
The matrix Р describes transition of the Jones vector from the Cartesian to spiral P-basis, where the relevant Jones vector will be denoted by the index , and the matrix N describes transition to the spiral N-basis with the index . If the neutral semi-axis is consistent with the X-axis of the Cartesian basis, then we have:
, . (2)
Transformation of the Jones matrix from the Cartesian polarization basis to the matrix in the spiral basis and vice versa is performed as follows:
. (3)
The polarization structures of the RP beam and the AP beam are intrinsic to the spiral basis, in which the vector is rotated counterclockwise when the azimuth angle ϕ is changed, while the polarization structure does not change when the coordinate axes are rotated.
When transforming from the Cartesian basis to the spiral -basis, the Jones vectors of these beams acquire the following form:
(4)
The polarization structures with a clockwise rotation of the vector oscillation plane are intrinsic for the N-basis. In this case, the polarization structure is changed when the Cartesian basis is rotated.
The Jones vectors of the RTP beam and LTP beam in a spiral basis have the following form:
(5)
3. Radial polarizer and devices on its basis
A necessary device for implementing the quantum key transmission, namely for identifying various basic states, is a radial polarizer (RP). The radial polarizer transmission axes are directed along the transverse radius . In the azimuth direction, i. e. when the vector is oriented tangent to the concentric circles, the transmission is equal to zero. We will show how the RP can be obtained using a conventional linear polarizer located between two spiral rotators [23].
The Jones matrices of the positive P and negative N spiral rotator in the Cartesian basis have the following form:
(6)
where ϕ is the azimuth angle measured from the horizontal X-axis of the Cartesian basis in the cross section of the beam; α is the angle made by the neutral semi-axis of the spiral rotator with the X-axis.
The spiral rotators can be polar or non-polar, depending on whether they change their sign for a reverse wave.
Let the neutral semi-axis of two spiral rotators with various signs coincide with the X-axis of the Cartesian basis. Let us place an ideal linear polarizer between these two rotators so that the axis of its greatest transmission makes an angle β with the Х axis. The Jones matrix of this polarization-inhomogeneous device will have the following form:
(7)
When β = 0, this polarization device is a radial polarizer that transmits the RP-beam without any losses, the vector of which is oriented along the transverse radius (Jones vector ) and completely absorbs the AP beam with an azimuth polarization structure (Jones vector ).
A radial polarizer generates a beam with a radial polarization structure from the linearly polarized light, regardless of the orientation of the vector oscillation plane – polarization azimuth , however, the beam intensity is changed:
(8)
where the Jones vector of linearly polarized light is described as follows:
In this case, the losses reach 50%.
The radial polarizer generates an optical vortex with a radial polarization structure based on the circularly polarized light:
(9)
In this case the losses are equal to 50%.
If we rearrange the spiral rotators in (5), then we obtain a hyperbolic polarizer, the eigenstates of polarization of which are the Jones vectors, in which, unlike the RP beam and the AP beam, the vector is rotated clockwise as the azimuth angle increases:
(10)
If polar Faraday spiral rotators are used in (5), then this device will be a radial polarizer for one direction and a hyperbolic polarizer for the opposite direction. Accordingly, will pass in one direction without any losses, and – in the other direction.
The Jones matrix of a radial polarizer in a spiral P-basis has the following form:
. (11)
Accordingly, the Jones matrix of the hyperbolic polarizer has a similar form in the spiral N-basis.
The radial polarizer, rotated by β, has the following form in a spiral basis:
(12)
In addition to the radial polarizer, three more devices are required to analyze the cryptographic key. Firstly, it is a radial polarizer, rotated by 90°, or, in other words, an axial polarizer that transmits the AP beam without any losses. The Jones matrix of this device in a spiral basis has the following form:
,
that corresponds to the substitution in (12) β = 90°.
The radial polarizer rotated by 45° is described by the Jones matrix in a spiral basis:
. (13)
This polarization device completely transmits the twisted RTP beam and completely quenches the orthogonally polarized LTP beam (4). For the other two polarization structures (3), the transmission is partial.
It should be noted that if we use the device consisting of a linear polarizer and two spiral rotators (7), then rotation of the linear polarizer by an angle of ±45° will lead to two devices for transmitting the RTP beam and the LTP beam.
The combined effect of four polarization inhomogeneous devices on the basic polarization states of the modified quantum key is shown in Table 2.
4. Description of protocol
The position of the vector oscillation plane at the output of the optical laser system varies over a wide range. It creates problems when determining the orientation of the reference coordinate system during the polarization key transmission from the ground abord and back. The use of polarization-symmetric structures allows to solve this problem.
The laser communication system with quantum cryptography according to the proposed protocol generates four photons on the basis of four types of axially symmetric beams shown in Table 2 using four devices with a radial polarizer in four orientations. The same four polarizing devices are used in the receiver portion, accordingly.
The first polarization device is a radial polarizer that quenches the AP beam and transmits the RP beam. The following matrix is obtained in the spiral basis:
(14)
The second polarization device is a radial polarizer rotated by 90° that quenches the RP beam and transmits the AP beam. These impacts are described as follows:
(15)
The third polarization device is a radial polarizer rotated by 45° that quenches the LTP beam and transmits the RTP beam. The following matrices are available in the spiral basis:
,
(16)
The fourth polarization device is a radial polarize rotated by –45° that quenches the RTP beam and transmits the LTP beam. The following is obtained:
(17)
Thus, we have shown that the use of Hermite-Gaussian beams with an axially symmetric polarization structure is similar to the well-known BB84 protocol in a spiral basis. However, the proposed practical implementation is invariant to the rotation relative to the propagation axis z.
The practical implementation can be performed as follows. The narrow-band laser and a quarter-wave plate shall be located at the input of the optical-laser guidance system for generating the circularly polarized light. Regardless of the telescope orientation of the transmitting system, such light shall retain its circular polarization. It is necessary to place a polarization device generating four basic beams with various axially symmetric polarization structures at the output of the guidance system. For example, it could be a radial polarizer between two spiral rotators that is rotated into four positions or a sorter. An important condition is coincidence of the optical axis of the beam and the radial polarizer. This can be achieved by using a guidance system with a coaxial beam having the sufficient power and possibly a different wavelength, for example, 532 nm, that is convenient for the system alignment. It should be noted that, in addition to the axially symmetric polarization structure, the output beams will have the properties of an optical vortex, since, according to (9), the phase beam structure is changed proportionally to the azimuth angle. As it is known [24, etc.], the optical vortices are more resistant to the atmospheric fluctuations than a conventional laser beam.
If the laser initially generates an RP beam, then it is sufficient to apply a polarization rotator, for example, a Faraday rotator that twists the polarization structure of the RP beam into an AP beam, a RTP beam, or an LTP beam.
The spacecraft shall have a response device in the form of a rotating radial polarizer and a relevant transponder.
As an illustration of the possible generation of beams considered in this paper, Fig. 2 shows the experimentally obtained images of the cross section of a beam passing through a turbulent medium, as well as a fragment of the interference pattern with a “fork” image that confirms the presence of an optical vortex.
Discussion of results, conclusions
The implementation of the BB84 protocol proposed in this paper using beams with axial symmetry of the polarization state is invariant with respect to rotation relative to the beam propagation axis, which makes it stable for the case of even a significant change in the polarization state at various points of the celestial hemisphere. Such a problem is typical for quantum key distribution systems in space via low-orbit spacecraft.
The practical implementation of Hermite-Gauss beams with an axially symmetric polarization structure does not cause difficulties [16–19, 21]. In our opinion, the issue of beam detection in the cryptographic data transmission systems is rather critical. An important step would be the practical implementation of spiral polarization rotators, for example, with the liquid-crystal films, and, accordingly, radial polarizers.
There are also implementation possibilities for such devices based on “thin” graphene layers [25–27] or electrically controlled diffraction gratings [28], or refractive biconic axicon [29].
The conceptual issue of such system implementation is that the principle of transmission unitarity in the quantum light theory requires the absence of optical losses in the detection system, since it leads to the loss of the photon entanglement state. Therefore, in the strict sense, such devices as the sorters, diffraction gratings, etc. cannot be used in the systems with a “true” photon source. However, up to the present time, in most practical cases, the QKD systems have used a highly stable narrow-band laser as a photon source (as, for example, in [30, 31]) that has disallowed the issue of losses in the individual photon detection systems.
AUTHORS
A. L. Sokolov, Doctor of Technical Sciences, Prof., Head of Department, JSC “Research-and-production corporation “Precision system and Instruments” (RPC PSI), Moscow, Russia.
ORCID: 0000-0001-6164-7615
V. M. Petrov, Doctor of Physical and Mathematical Sciences (radiophysics), Doctor of Physical and Mathematical Sciences (optics), professor, Department of General Physics No. 1, Saint-Petersburg State University, Saint-Petersburg, Russia
ORCID: 0000-0002-8523-0336
V. Yu. Venediktov, Dr. of Sciences (Phys.&Math.), Professor, Chief Researcher, Department of Laser Measurement and Navigation Systems, St. Petersburg Electrotechnical University “LETI”, St. Petersburg, Russia.
ORCID: 0000-0055-1234-5678
D. D. Reshetnikov, Ph. D. Student, Department of General Physics No.1, Saint-Petersburg State University, Saint-Petersburg, Russia
CONFLICT OF INTEREST
The authors declare that they have no conflict of interest. The study results were discussed and indicated in the manuscript being a joint paper.
A. L. Sokolov 1, V. M. Petrov 2, V. Yu. Venediktov 2, D. D. Reshetnikov 2
JSC “Research-and-production corporation “Precision system and Instruments” (RPC PSI), Moscow, Russia
Saint-Petersburg State University, Saint-Petersburg, Russia
Department of Laser ³Measurement and Navigation Systems, St. Petersburg Electrotechnical University “LETI”, St. Petersburg, Russia
The paper provides a description of Hermite-Gaussian beams with an axially symmetric polarization structure generated by a vector superposition of Hermite-Gaussian modes with the indices 10 and 01. It is shown that the axial polarization symmetry makes such beams insensitive to the rotations relative to the optical axis that makes such a modification of the well-known BB84 protocol preferred for the quantum space cryptography systems. The possible creation and detection of such beams within the framework of a polarization protocol transmission using the devices with a radial polarizer is discussed.
Keywords: quantum cryptography, polarization protocol, axially symmetric polarization structure, radial polarizer, spiral rotator
Article received:04.09.2023
Article accepted:13.10.2023
Introduction
The development of quantum cryptography systems [1] is driven by the need to protect information in the up-to-date communication networks. The use of various types of spatially structured beams can be one of the efficient solutions to the problem of optical data transmission in real turbulence conditions, both through the classical and quantum communication channels.
At present, a large number of theoretical and experimental works has been performed in this area, and a large number of articles have been published. For example, the influence of atmospheric channel turbulence on the propagation of Laguerre-Gaussian and Bessel-Gaussian vector beams and on the channel capacity has been studied [2, 3]; the transmission of higher modes of Laguerre-Gaussian beams in the real conditions of urban turbulence over a distance of 1.6 km has been demonstrated using a wavelength of 809 nm [4]; a key distribution rate of at least 120 Mbit/s has been experimentally demonstrated using the spectral, polarization and orbital angular momentum multiplexing at a wavelength of 1550.12 nm (193.4 THz) [5]. In the last example, the simulation of atmospheric turbulence has been performed using two two-dimensional phase light modulators, providing the turbulence corresponding to the value of the Rytov parameter . The quantum key distribution through a vortex optical fiber with the length of 60 m has been experimentally confirmed. In this case, a pair of entangled photons has been used, obtained as a result of the spontaneous parametric scattering at a wavelength of 405 nm [6]. The works are actively performed to study the propagation of Gaussian and vortex beams under the atmospheric turbulence conditions over the significant distances, up to 1 000 m [7].
A special place is held by the data transmission issues over distances of more than a thousand kilometers using the low-orbiting spacecrafts with the appropriate equipment [8–11]. If back in 2017, the paper [9] reported the achieved key transmission speed of “several kilohertz” to a satellite located at a distance of 1 200 km from the Earth, then in 2021 it was already reported about an “integrated fiber-optic and satellite network” with a total length 4 600 km and the secret key transmission to the satellite at a speed of 47.8 kBit/s [10]. It shall be noted that the possible application of beams with an axially symmetric polarization structure in the space systems, i. e., having the spatial polarization modulation in a plane orthogonal to the propagation direction, as the base beams that form a polarization cryptographic key is shown in the paper [11].
Almost all well-known papers devoted to the quantum key distribution (QKD) apply the BB84 protocol that uses two bases, each of which contains two photon states. In the first basis, the photon is linearly polarized in a vertical or horizontal way (0° or 90°); in the second basis, the photon is linearly polarized in a diagonal way (45° or 135°). In particular, this is due to the instability of the “phase” protocols during the propagation of light in a turbulent atmosphere.
At the same time, it should be noted that the application of the BB84 protocol using bases based on linear polarization of photons for the tasks of low-orbit spacecraft has its difficulties associated with the need to fix the position of the plane of polarization of light at each moment of time, both transmitting and receiving systems on earth and in space. The analysis shows that in the transmitting optical and laser systems, the polarization condition is changed significantly for different points of the hemisphere [12]. In the case of the polarization protocol, this means the dependence of two coordinate systems rotated by 45° on the relative orientation of the transmitting telescope and the spacecraft.
This dependence can be eliminated if the beams with an axially symmetric polarization structure are used [13–15].
The separate tasks are both obtaining the beams with a given axial polarization structure and their detection. Their generation methods can be divided into two main ones: the first group include the intracavity methods, when the first-order modes are generated instead of the main laser mode [16, 17], and the extracavity methods using the diffractive optical elements [18]. The paper [19] has shown that the second-order beams are generated when a linearly polarized beam is reflected from an angle reflector [20, 21]. In particular, a second-order optical vortex is generated in the presence of a special interference coating of the faces. The beam detection issue can be solved either by using a device that acts as a radial polarizer [15], or by using a device similar to the diffraction sorter, for example, [22].
The purpose of this paper is to propose an implementation of the well-known BB84 protocol for a space-based quantum data transmission system. A feature of such an implementation shall be its invariance in relation to the rotation around the z axis coinciding with the beam propagation direction.
1. Beams with an axially symmetric polarization structure
In the case of an axially symmetric polarization structure, regardless of the radial coordinate in the beam cross-sectional plane, each azimuth value corresponds to a certain orientation of the vector oscillation plane that is changed so that when returning to the original azimuth value, this plane makes an integer number of revolutions. The polarization-symmetric structures have two modifications depending on the rotation direction of the vector oscillation plane .
The polarization structure of these beams is invariant to the rotation relative to the beam axis: the polarization condition is maintained along the radius vector r for an arbitrary azimuth angle ϕ (Fig. 1).
In this paper, the beams with an axisymmetric polarization structure are proposed to be used when transmitting the quantum keys in outer space using the BB84 protocol.
Table 1 shows how the axially symmetric structures that make up the set for the modified polarization protocol are generated.
The basic orthogonal polarization structures of the polarization protocol are the beams formed by a vector superposition of linearly polarized Hermite-Gaussian modes with the indices 10 and 01 (Table 1). This is a radial polarization structure, since the vector at each point of the transverse plane is oriented along the radius (RP-beam), and an azimuth polarization structure, since the vector is directed at each point at a tangent to the concentric circles (AP-beam), and two orthogonal polarization structures with axial symmetry that are rotated at 45° relative to the RP-beam and AP-beam: right-twisted (RTP-beam) and left-twisted (LTP-beam). The Jones vectors in Table 1 are recorded in a cylindrical basis, where ϕ is the azimuth angle.
It is convenient to demonstrate the interaction of basic beams with the radial polarization elements in a special polarization (spiral) basis.
2. Spiral bases
For the convenience of mathematical manipulations, the spiral bases that are specified using two matrices [8] are used:
(1)
where ϕ is the azimuth angle, measured from the so-called neutral semi-axis, in the direction of which the matrices become unitary.
The matrix Р describes transition of the Jones vector from the Cartesian to spiral P-basis, where the relevant Jones vector will be denoted by the index , and the matrix N describes transition to the spiral N-basis with the index . If the neutral semi-axis is consistent with the X-axis of the Cartesian basis, then we have:
, . (2)
Transformation of the Jones matrix from the Cartesian polarization basis to the matrix in the spiral basis and vice versa is performed as follows:
. (3)
The polarization structures of the RP beam and the AP beam are intrinsic to the spiral basis, in which the vector is rotated counterclockwise when the azimuth angle ϕ is changed, while the polarization structure does not change when the coordinate axes are rotated.
When transforming from the Cartesian basis to the spiral -basis, the Jones vectors of these beams acquire the following form:
(4)
The polarization structures with a clockwise rotation of the vector oscillation plane are intrinsic for the N-basis. In this case, the polarization structure is changed when the Cartesian basis is rotated.
The Jones vectors of the RTP beam and LTP beam in a spiral basis have the following form:
(5)
3. Radial polarizer and devices on its basis
A necessary device for implementing the quantum key transmission, namely for identifying various basic states, is a radial polarizer (RP). The radial polarizer transmission axes are directed along the transverse radius . In the azimuth direction, i. e. when the vector is oriented tangent to the concentric circles, the transmission is equal to zero. We will show how the RP can be obtained using a conventional linear polarizer located between two spiral rotators [23].
The Jones matrices of the positive P and negative N spiral rotator in the Cartesian basis have the following form:
(6)
where ϕ is the azimuth angle measured from the horizontal X-axis of the Cartesian basis in the cross section of the beam; α is the angle made by the neutral semi-axis of the spiral rotator with the X-axis.
The spiral rotators can be polar or non-polar, depending on whether they change their sign for a reverse wave.
Let the neutral semi-axis of two spiral rotators with various signs coincide with the X-axis of the Cartesian basis. Let us place an ideal linear polarizer between these two rotators so that the axis of its greatest transmission makes an angle β with the Х axis. The Jones matrix of this polarization-inhomogeneous device will have the following form:
(7)
When β = 0, this polarization device is a radial polarizer that transmits the RP-beam without any losses, the vector of which is oriented along the transverse radius (Jones vector ) and completely absorbs the AP beam with an azimuth polarization structure (Jones vector ).
A radial polarizer generates a beam with a radial polarization structure from the linearly polarized light, regardless of the orientation of the vector oscillation plane – polarization azimuth , however, the beam intensity is changed:
(8)
where the Jones vector of linearly polarized light is described as follows:
In this case, the losses reach 50%.
The radial polarizer generates an optical vortex with a radial polarization structure based on the circularly polarized light:
(9)
In this case the losses are equal to 50%.
If we rearrange the spiral rotators in (5), then we obtain a hyperbolic polarizer, the eigenstates of polarization of which are the Jones vectors, in which, unlike the RP beam and the AP beam, the vector is rotated clockwise as the azimuth angle increases:
(10)
If polar Faraday spiral rotators are used in (5), then this device will be a radial polarizer for one direction and a hyperbolic polarizer for the opposite direction. Accordingly, will pass in one direction without any losses, and – in the other direction.
The Jones matrix of a radial polarizer in a spiral P-basis has the following form:
. (11)
Accordingly, the Jones matrix of the hyperbolic polarizer has a similar form in the spiral N-basis.
The radial polarizer, rotated by β, has the following form in a spiral basis:
(12)
In addition to the radial polarizer, three more devices are required to analyze the cryptographic key. Firstly, it is a radial polarizer, rotated by 90°, or, in other words, an axial polarizer that transmits the AP beam without any losses. The Jones matrix of this device in a spiral basis has the following form:
,
that corresponds to the substitution in (12) β = 90°.
The radial polarizer rotated by 45° is described by the Jones matrix in a spiral basis:
. (13)
This polarization device completely transmits the twisted RTP beam and completely quenches the orthogonally polarized LTP beam (4). For the other two polarization structures (3), the transmission is partial.
It should be noted that if we use the device consisting of a linear polarizer and two spiral rotators (7), then rotation of the linear polarizer by an angle of ±45° will lead to two devices for transmitting the RTP beam and the LTP beam.
The combined effect of four polarization inhomogeneous devices on the basic polarization states of the modified quantum key is shown in Table 2.
4. Description of protocol
The position of the vector oscillation plane at the output of the optical laser system varies over a wide range. It creates problems when determining the orientation of the reference coordinate system during the polarization key transmission from the ground abord and back. The use of polarization-symmetric structures allows to solve this problem.
The laser communication system with quantum cryptography according to the proposed protocol generates four photons on the basis of four types of axially symmetric beams shown in Table 2 using four devices with a radial polarizer in four orientations. The same four polarizing devices are used in the receiver portion, accordingly.
The first polarization device is a radial polarizer that quenches the AP beam and transmits the RP beam. The following matrix is obtained in the spiral basis:
(14)
The second polarization device is a radial polarizer rotated by 90° that quenches the RP beam and transmits the AP beam. These impacts are described as follows:
(15)
The third polarization device is a radial polarizer rotated by 45° that quenches the LTP beam and transmits the RTP beam. The following matrices are available in the spiral basis:
,
(16)
The fourth polarization device is a radial polarize rotated by –45° that quenches the RTP beam and transmits the LTP beam. The following is obtained:
(17)
Thus, we have shown that the use of Hermite-Gaussian beams with an axially symmetric polarization structure is similar to the well-known BB84 protocol in a spiral basis. However, the proposed practical implementation is invariant to the rotation relative to the propagation axis z.
The practical implementation can be performed as follows. The narrow-band laser and a quarter-wave plate shall be located at the input of the optical-laser guidance system for generating the circularly polarized light. Regardless of the telescope orientation of the transmitting system, such light shall retain its circular polarization. It is necessary to place a polarization device generating four basic beams with various axially symmetric polarization structures at the output of the guidance system. For example, it could be a radial polarizer between two spiral rotators that is rotated into four positions or a sorter. An important condition is coincidence of the optical axis of the beam and the radial polarizer. This can be achieved by using a guidance system with a coaxial beam having the sufficient power and possibly a different wavelength, for example, 532 nm, that is convenient for the system alignment. It should be noted that, in addition to the axially symmetric polarization structure, the output beams will have the properties of an optical vortex, since, according to (9), the phase beam structure is changed proportionally to the azimuth angle. As it is known [24, etc.], the optical vortices are more resistant to the atmospheric fluctuations than a conventional laser beam.
If the laser initially generates an RP beam, then it is sufficient to apply a polarization rotator, for example, a Faraday rotator that twists the polarization structure of the RP beam into an AP beam, a RTP beam, or an LTP beam.
The spacecraft shall have a response device in the form of a rotating radial polarizer and a relevant transponder.
As an illustration of the possible generation of beams considered in this paper, Fig. 2 shows the experimentally obtained images of the cross section of a beam passing through a turbulent medium, as well as a fragment of the interference pattern with a “fork” image that confirms the presence of an optical vortex.
Discussion of results, conclusions
The implementation of the BB84 protocol proposed in this paper using beams with axial symmetry of the polarization state is invariant with respect to rotation relative to the beam propagation axis, which makes it stable for the case of even a significant change in the polarization state at various points of the celestial hemisphere. Such a problem is typical for quantum key distribution systems in space via low-orbit spacecraft.
The practical implementation of Hermite-Gauss beams with an axially symmetric polarization structure does not cause difficulties [16–19, 21]. In our opinion, the issue of beam detection in the cryptographic data transmission systems is rather critical. An important step would be the practical implementation of spiral polarization rotators, for example, with the liquid-crystal films, and, accordingly, radial polarizers.
There are also implementation possibilities for such devices based on “thin” graphene layers [25–27] or electrically controlled diffraction gratings [28], or refractive biconic axicon [29].
The conceptual issue of such system implementation is that the principle of transmission unitarity in the quantum light theory requires the absence of optical losses in the detection system, since it leads to the loss of the photon entanglement state. Therefore, in the strict sense, such devices as the sorters, diffraction gratings, etc. cannot be used in the systems with a “true” photon source. However, up to the present time, in most practical cases, the QKD systems have used a highly stable narrow-band laser as a photon source (as, for example, in [30, 31]) that has disallowed the issue of losses in the individual photon detection systems.
AUTHORS
A. L. Sokolov, Doctor of Technical Sciences, Prof., Head of Department, JSC “Research-and-production corporation “Precision system and Instruments” (RPC PSI), Moscow, Russia.
ORCID: 0000-0001-6164-7615
V. M. Petrov, Doctor of Physical and Mathematical Sciences (radiophysics), Doctor of Physical and Mathematical Sciences (optics), professor, Department of General Physics No. 1, Saint-Petersburg State University, Saint-Petersburg, Russia
ORCID: 0000-0002-8523-0336
V. Yu. Venediktov, Dr. of Sciences (Phys.&Math.), Professor, Chief Researcher, Department of Laser Measurement and Navigation Systems, St. Petersburg Electrotechnical University “LETI”, St. Petersburg, Russia.
ORCID: 0000-0055-1234-5678
D. D. Reshetnikov, Ph. D. Student, Department of General Physics No.1, Saint-Petersburg State University, Saint-Petersburg, Russia
CONFLICT OF INTEREST
The authors declare that they have no conflict of interest. The study results were discussed and indicated in the manuscript being a joint paper.
Readers feedback
