rus
DOI: 10.22184/1993-7296.FRos.2025.19.5.378.388
This article presents the development of a computer model for calculating the optical specifications of a liquid crystal modulator. It allows considering multipath interference in the device with acceptable approximations and does not require any significant computational costs. The performance analysis of a THz radiation modulator based on a multicellular liquid crystal structure is performed.
This article presents the development of a computer model for calculating the optical specifications of a liquid crystal modulator. It allows considering multipath interference in the device with acceptable approximations and does not require any significant computational costs. The performance analysis of a THz radiation modulator based on a multicellular liquid crystal structure is performed.
Теги: liquid crystal structure thz radiation modulator жидкокристаллическая структура модулятор тгц-излучения
π-Cell Based Liquid Crystal Modulator for THz Measurements
G. V. Simonenko
Chernyshevsky Saratov National Research State University, Saratov, Russia
This article presents the development of a computer model for calculating the optical specifications of a liquid crystal modulator. It allows considering multipath interference in the device with acceptable approximations and does not require any significant computational costs. The performance analysis of a THz radiation modulator based on a multicellular liquid crystal structure is performed.
Key words: THz radiation modulator, liquid crystal structure
Article received:07.03.2025
Article accepted:29.04.2025
Introduction
Recently, interest in the terahertz (THz) range of electromagnetic radiation has increased in terms of its practical use in the technical and medical applications [1, 2]. One of the main elements of terahertz technology is a modulator of such radiation, and the most promising device for this purpose is a liquid crystal (LC) device [3, 4]. The standard research tool for the new design of an LC modulator is computer modeling based on various matrix optics methods [5, 6, 7]. The Jones and Berreman matrix methods are most often used to calculate the performance of data displaying and processing LC devices [5, 6]. It should be noted that when the LC devices are operated in the THz range, a significant role is played by the multipath interference phenomenon [6, 8] that cannot be considered within the framework of the Jones matrix method [5]. On the other part, the Berreman matrix method is used to take this phenomenon into account in the LC devices, the application of which requires great computational costs [8]. In this regard, a critical task is to develop a simple computer model for calculating the optical specifications of an LC modulator that would allow to consider multi-wave interference in such a device with acceptable approximations and would not require any significant computational costs. The solution to this problem is given in this article. The performance analysis of a THz radiation modulator based on a multicellular LC structure is conducted on its basis.
Optical performance simulation of the liquid crystal modulator of electromagnetic radiation
In terms of the optimal ratio between the optical (maximum transmission and contrast ratio) and dynamic (minimum response and relaxation times) specifications of the LC modulator, the best design is a device based on a typical π-cell [9, 10]. If the modulated radiation is not polarized, the LC cell is placed between two crossed polarizers. The orientation angles of the polarizers in this case are ±45° with the LC optical axis. If the initially modulated radiation is linearly polarized, the input polarizer is absent and the angle between the polarization vector direction of this radiation is 45° with the LC optical axis, and the output polarizer is placed at an angle of −45° to this direction. In this case, the radiation is modulated due to the effect of electric field-controlled birefringence [5, 10]. The LC cell control mode shall be as follows: maximum transmission of the device corresponds to the zero value of the control voltage; the minimum transmission value corresponds to a voltage with a frequency of 1000 Hz and an amplitude of 15–20 V. At intermediate values of the control voltage, the modulator’s transmission is changed from the maximum to the minimum level.
Most often, such devices are used when the following conditions are met: 1) the modulated radiation is propagated parallel to the normal to the modulator surface [5]; 2) the modulator base is a homogeneous LC cell filled with a nematic material with antisymmetric boundary conditions and with a zero twist angle of the LC structure [5]. The antisymmetric boundary conditions in the LC cell mean that the tilt angle of the LC molecules on one orienting surface has the same value as the tilt angle of the LC molecules on the other orienting surface, but with the opposite sign.
Under these conditions, the LC modulator transmission TM can be calculated using the following well-known expression [11]:
TM = Tinter · sin2 , (1)
where llc – thickness of the LC layer; Δnlc = ne – no; no – an ordinary refractive index of the LC layer; ne – an extraordinary refractive index of the LC layer with the average thickness at a certain control voltage value; λ – wavelength of the modulated radiation; Tinter – transmission coefficient which value depends on the losses due to the multi-wave interference in the multilayer planar structure of the LC cell (0 ≤ Tinter ≤ 1).
The value of the LC extraordinary refractive index with the average thickness is determined by the distribution function of the inclination angle of the LC molecules θ(x) to the orienting surface (x is the current orientation coordinate of the LC director over its thickness). The distribution function of the orientation angle of the LC director θ(x) depends on the control electric field voltage, on the values of the physical LC constants and on the boundary conditions on the LC cell orienting surfaces. This function is determined by solving the deformation problem [5–7], and in our case it is considered well-known.
Fig. 1 schematically shows the design of a classical LC π-cell (Fig. 1). Let a linearly polarized electromagnetic wave fall on such a structure at an incidence angle of 0°. Let us represent the original electromagnetic wave as the sum of an ordinary (o-type) and extraordinary (e-type) waves. In this case, the polarization vectors for the o- and e-types are perpendicular to each other. The polarization type of the wave propagating in the LC cell is unchanged, since the radiation propagates along the normal to the cell surface. Therefore, the transmission coefficient for the ordinary wave Tointer in such a system, and then the transmission coefficient for the extraordinary wave Teinter can be calculated independently of each other. Then the transmission coefficient of the total radiation can be obtained by incoherent addition of the transmission coefficients for the ordinary and extraordinary waves. To solve this problem, the Abeles matrix formalism [12] is applied, using the ordinary and extraordinary wave polarizations as the basic polarizations. In this case, the device will be described by two complex matrices 2 × 2 So and Se (for the ordinary and extraordinary waves, respectively). Each of the matrices So and Se can be represented as the products of interface matrices and layer matrices, describing the influence of individual layers and interfaces in the entire layered structure [12].
Then the amplitude transmission coefficient for an ordinary (extraordinary) electromagnetic wave τо(е)and the relevant transmission coefficient Tо(е)inter are calculated using the matrix element S11o(e) of the total matrix of the device So(Se) as follows:
τо(е) = 1S11o(e), Tо(е)inter = τо(е) · τ*о(е). (2)
If a linearly polarized electromagnetic wave falls on an LC cell and the angle between the polarization direction and the LC optical axis is equal to α, then the total transmission coefficient (energy transmittance) for the electromagnetic wave Tinter is determined using the following expression:
Tinter = Tointer cos2 α + Teinter sin2 α. (3)
In the case, when the unpolarized electromagnetic radiation falls on the LC cell or the angle α is equal to 45°, then the expression (3) can be rewritten as follows:
Tinter = Tointer + Teinter2. (4)
While using the procedure described above, it is possible to easily calculate the transmission coefficient of electromagnetic radiation for the modulator Tinter, operating on the basis of the typical π-cell. The calculation algorithm for the Tinter coefficient of the LC modulator in this case consists of the following stages. At the first stage, the distribution of the orientation angles of the LC director θ(x) is calculated for a given control voltage with due regard to the boundary conditions in the LC cell and the LC physical constants (elasticity coefficients and dielectric constants). This problem is not included in the calculation part of the proposed method, but is taken from a well-known software package [13]. At the second stage, it is possible to calculate the transmission coefficient of the ordinary wave Tointer while assuming that the LC layer is described by the ordinary refractive index no. At the third stage, we consider that the LC layer is described by the refractive index with the average layer thickness for the extraordinary wave ne and calculate the transmission coefficient of the extraordinary wave Teinter. If a linearly polarized electromagnetic wave falls on the LC cell which polarization direction makes an angle α with the optical LC axis, then the total transmission coefficient Tinter for the electromagnetic wave is calculated using the expression (3). When the unpolarized electromagnetic radiation falls on the LC cell, the energy transmission coefficient Tinter is calculated using the formula (4). At the fourth stage, the transmission coefficient of the entire LC modulator of electromagnetic radiation TM is calculated using the expressions (1).
On the basis of the described algorithm, the programm lc the Fortran 95 language was developed for calculation of energy transmission coefficient of the LCD modulator operating on the basis of the typical π-cell [10]. The input parameters of the lc program are the following parameters of the LC cell: refractive indices of the medium (n1, n9) where the cell is located; refractive index and thickness of the glass substrates (n2, n8, l2, l8); refractive index and thickness of the electrode layers (n3, n7, l3, l7); refractive index and thickness of the orienting layers (n4, n6, l4, l6); refractive indices of the LC (no, ne) and thickness of its layer llc. In addition, the input program parameter is the wavelength of the modulated electromagnetic radiation λ and the initial distribution of orientation angles of the LC director θ(x), as well as the control modulator voltage. The output parameter of this program is the transmission coefficient value of the entire LC modulator TM.
Results and discussion
To test the proposed calculation method for the transmission coefficient of the LC modulator, a comparison was made of the simulation results related to the device specifications obtained using the described method with the results obtained using the well-known MOUSE – LCD software package [13]. The physical parameters of the LC mixture and the process parameters of the LC cell that have been used in the calculations, are given in [9, 14]. The calculations were performed for a radiation with a wavelength λ = 0.5 μm.
Fig. 2 shows the simulation results of the dependence of the LC modulator transmission coefficient TM on the control voltage U, obtained using the proposed method and the MOUSE-LCD software package. The discrepancy between these data does not exceed 10% that corresponds to the calculation accuracy of the MOUSE-LCD software package and indicates the adequacy of the proposed simple method for calculating the transmission coefficient of the LC modulator based on the π-cell.
One of the main tasks in designing the LC modulators is to minimize the losses related to the available Fresnel losses due to reflection from the plane-parallel layers with various refractive indices [15]. In this case, as a rule, the dependence of these losses is considered only in relation to the design parameters of the LC cell (see, for example, [10]), and the influence of the control voltage is not taken into account. However, such losses depend on the nature of the multi-wave interference in the layered LC cell structure, and, therefore, also depend on the extraordinary refractive LC index which value is changed as the control voltage evolves. Therefore, the issue of influence of the control LC cell voltage on the losses related to the multipath interference in the layered cell structure is of great interest. Figure 3 shows the dependences of the transmission coefficient of an LC cell without any polarizers on the control voltage. It is clear that the Fresnel reflection losses with due regard to the multi-wave interference, depends quite strongly on the control voltage and the modulated radiation wavelength. The difference between the loss values, for example, for the wavelengths of 0.5 μm and 16 μm can reach 50%. The dependence nature is also determined by the wavelength of electromagnetic radiation passing through the LC cell. Thus, for the radiation wavelengths of 0.5, 1, 2 and 8 μm, the transmission coefficient Tinter with an increased control voltage U has the shape of a curve with a minimum, and for the wavelengths of 4 and 16 μm, the dependence Tinter(U) has the shape of a curve with a maximum. The value of changes can be about 15–20% of the maximum value while indicating the possible modulation of electromagnetic radiation even in a polaroid-free modulator structure.
However, in this case, the amplitude modulation coefficient m does not exceed 11%. Two conclusions can be drawn:
the value of Fresnel losses in the LC cell can be adjusted by selecting the control voltage in the condition with maximum transmission;
if the modulation of electromagnetic radiation is required within a wide range of wavelengths, but with a low amplitude modulation coefficient, then only one non-polaroid LC cell can be used for this purpose.
To obtain higher values of the amplitude modulation coefficient, it is necessary to use a modulator design with polaroids. In this case, for THz radiation, the optimal thickness of the working clearance of the LC cell llc is determined based on the maximum interference of polarized waves [4]:
λ0 = 2 · ne − no · llc, (5)
where λ0 is the modulated radiation wavelength.
In this case, the standard structure of the THz radiation LC modulator based on only one π-cell has one significant drawback: it has too long full response time [4]. Previously, a more comprehensive design of the LC modulator based on N identical elementary π-cells was proposed to solve this issue. Each of these elementary cells has its own electric control, and the entire set of such cells is placed between two polarizers. The number of such π-cells N is determined by the condition of maximum interference of the polarized waves [4]:
λ0 = 2 · N · ne − no · l0, (6)
where L0 is the thickness of one elementary π-cell.
The table demonstrates the results of computer simulation of the specifications of THz-radiation LC modulators for various wavelengths. It should be noted that each LC modulator is designed only for a certain wavelength and has a different number of elementary LC cells. The following set of specifications was selected to describe the LC modulator, such as an amplitude modulation coefficient m, maximum open-state transmission coefficient Tmax and total response time τ. The modulators differ in the number N of elementary π-cells. It follows from the table that the LC modulators for various wavelengths may differ insignificantly only in the maximum transmittance value, with all other specifications being equal. This is explained by the fact that all modulators are based on the same elementary LC cell and satisfy the polarized wave interference condition (5). The difference in the maximum transmittance values is resulted from the losses in multilayer structures with the increased interfaces.
Conclusion
This paper proposes a simple matrix calculation method for the energy transmission coefficient of the LC modulator based on a typical π-cell. The method is based on the matrix formalism of comprehensive 2 × 2 Abeles matrices and allows for simultaneous consideration of the multipath interference in the layered cell structure and the phenomenon of electric field-controlled birefringence in the LC. Comparison of the data obtained using the method described above with the calculation results obtained by the proven software packages has shown their good quantitative fit. This allows us to confirm the adequacy and practical suitability of the proposed method to simulate the specifications of the LC modulator based on a π-cell. The simulation results have showed that the losses in the LC cell that are caused by the Fresnel reflections from the cell structure interfaces and multi-wave interference in this structure, depends on the control voltage and is different for various wavelengths of the electromagnetic range. The difference between the loss value, for example, for the wavelengths of 0.5 μm and 16 μm, can reach 50%.
In addition, dependence of the energy transmission coefficient of the LC cell on the control electric field can be used to modulate the electromagnetic radiation of various ranges with a small value of the amplitude modulation coefficient. It has been shown by the computer method that the application of the LC modulator structure consisting of a certain number of identical π-cells allows achieving acceptable parameters of the modulated THz radiation. In this case, the number of elementary LC cells is determined by the maximum interference of polarized waves, wavelength of the modulated THz radiation and thickness of the elementary LC cell.
AUTHOR
Simonenko Georgy V., Dr. of Sc. (Phys.&Math.), Prof. Department of Optics and Biophotonics, Chernyshevsky Saratov National Research State University, e-mail: simonenkogv@sgu.ru; Saratov, Russia.
ORCID: 0000-0002-6283-6335
G. V. Simonenko
Chernyshevsky Saratov National Research State University, Saratov, Russia
This article presents the development of a computer model for calculating the optical specifications of a liquid crystal modulator. It allows considering multipath interference in the device with acceptable approximations and does not require any significant computational costs. The performance analysis of a THz radiation modulator based on a multicellular liquid crystal structure is performed.
Key words: THz radiation modulator, liquid crystal structure
Article received:07.03.2025
Article accepted:29.04.2025
Introduction
Recently, interest in the terahertz (THz) range of electromagnetic radiation has increased in terms of its practical use in the technical and medical applications [1, 2]. One of the main elements of terahertz technology is a modulator of such radiation, and the most promising device for this purpose is a liquid crystal (LC) device [3, 4]. The standard research tool for the new design of an LC modulator is computer modeling based on various matrix optics methods [5, 6, 7]. The Jones and Berreman matrix methods are most often used to calculate the performance of data displaying and processing LC devices [5, 6]. It should be noted that when the LC devices are operated in the THz range, a significant role is played by the multipath interference phenomenon [6, 8] that cannot be considered within the framework of the Jones matrix method [5]. On the other part, the Berreman matrix method is used to take this phenomenon into account in the LC devices, the application of which requires great computational costs [8]. In this regard, a critical task is to develop a simple computer model for calculating the optical specifications of an LC modulator that would allow to consider multi-wave interference in such a device with acceptable approximations and would not require any significant computational costs. The solution to this problem is given in this article. The performance analysis of a THz radiation modulator based on a multicellular LC structure is conducted on its basis.
Optical performance simulation of the liquid crystal modulator of electromagnetic radiation
In terms of the optimal ratio between the optical (maximum transmission and contrast ratio) and dynamic (minimum response and relaxation times) specifications of the LC modulator, the best design is a device based on a typical π-cell [9, 10]. If the modulated radiation is not polarized, the LC cell is placed between two crossed polarizers. The orientation angles of the polarizers in this case are ±45° with the LC optical axis. If the initially modulated radiation is linearly polarized, the input polarizer is absent and the angle between the polarization vector direction of this radiation is 45° with the LC optical axis, and the output polarizer is placed at an angle of −45° to this direction. In this case, the radiation is modulated due to the effect of electric field-controlled birefringence [5, 10]. The LC cell control mode shall be as follows: maximum transmission of the device corresponds to the zero value of the control voltage; the minimum transmission value corresponds to a voltage with a frequency of 1000 Hz and an amplitude of 15–20 V. At intermediate values of the control voltage, the modulator’s transmission is changed from the maximum to the minimum level.
Most often, such devices are used when the following conditions are met: 1) the modulated radiation is propagated parallel to the normal to the modulator surface [5]; 2) the modulator base is a homogeneous LC cell filled with a nematic material with antisymmetric boundary conditions and with a zero twist angle of the LC structure [5]. The antisymmetric boundary conditions in the LC cell mean that the tilt angle of the LC molecules on one orienting surface has the same value as the tilt angle of the LC molecules on the other orienting surface, but with the opposite sign.
Under these conditions, the LC modulator transmission TM can be calculated using the following well-known expression [11]:
TM = Tinter · sin2 , (1)
where llc – thickness of the LC layer; Δnlc = ne – no; no – an ordinary refractive index of the LC layer; ne – an extraordinary refractive index of the LC layer with the average thickness at a certain control voltage value; λ – wavelength of the modulated radiation; Tinter – transmission coefficient which value depends on the losses due to the multi-wave interference in the multilayer planar structure of the LC cell (0 ≤ Tinter ≤ 1).
The value of the LC extraordinary refractive index with the average thickness is determined by the distribution function of the inclination angle of the LC molecules θ(x) to the orienting surface (x is the current orientation coordinate of the LC director over its thickness). The distribution function of the orientation angle of the LC director θ(x) depends on the control electric field voltage, on the values of the physical LC constants and on the boundary conditions on the LC cell orienting surfaces. This function is determined by solving the deformation problem [5–7], and in our case it is considered well-known.
Fig. 1 schematically shows the design of a classical LC π-cell (Fig. 1). Let a linearly polarized electromagnetic wave fall on such a structure at an incidence angle of 0°. Let us represent the original electromagnetic wave as the sum of an ordinary (o-type) and extraordinary (e-type) waves. In this case, the polarization vectors for the o- and e-types are perpendicular to each other. The polarization type of the wave propagating in the LC cell is unchanged, since the radiation propagates along the normal to the cell surface. Therefore, the transmission coefficient for the ordinary wave Tointer in such a system, and then the transmission coefficient for the extraordinary wave Teinter can be calculated independently of each other. Then the transmission coefficient of the total radiation can be obtained by incoherent addition of the transmission coefficients for the ordinary and extraordinary waves. To solve this problem, the Abeles matrix formalism [12] is applied, using the ordinary and extraordinary wave polarizations as the basic polarizations. In this case, the device will be described by two complex matrices 2 × 2 So and Se (for the ordinary and extraordinary waves, respectively). Each of the matrices So and Se can be represented as the products of interface matrices and layer matrices, describing the influence of individual layers and interfaces in the entire layered structure [12].
Then the amplitude transmission coefficient for an ordinary (extraordinary) electromagnetic wave τо(е)and the relevant transmission coefficient Tо(е)inter are calculated using the matrix element S11o(e) of the total matrix of the device So(Se) as follows:
τо(е) = 1S11o(e), Tо(е)inter = τо(е) · τ*о(е). (2)
If a linearly polarized electromagnetic wave falls on an LC cell and the angle between the polarization direction and the LC optical axis is equal to α, then the total transmission coefficient (energy transmittance) for the electromagnetic wave Tinter is determined using the following expression:
Tinter = Tointer cos2 α + Teinter sin2 α. (3)
In the case, when the unpolarized electromagnetic radiation falls on the LC cell or the angle α is equal to 45°, then the expression (3) can be rewritten as follows:
Tinter = Tointer + Teinter2. (4)
While using the procedure described above, it is possible to easily calculate the transmission coefficient of electromagnetic radiation for the modulator Tinter, operating on the basis of the typical π-cell. The calculation algorithm for the Tinter coefficient of the LC modulator in this case consists of the following stages. At the first stage, the distribution of the orientation angles of the LC director θ(x) is calculated for a given control voltage with due regard to the boundary conditions in the LC cell and the LC physical constants (elasticity coefficients and dielectric constants). This problem is not included in the calculation part of the proposed method, but is taken from a well-known software package [13]. At the second stage, it is possible to calculate the transmission coefficient of the ordinary wave Tointer while assuming that the LC layer is described by the ordinary refractive index no. At the third stage, we consider that the LC layer is described by the refractive index with the average layer thickness for the extraordinary wave ne and calculate the transmission coefficient of the extraordinary wave Teinter. If a linearly polarized electromagnetic wave falls on the LC cell which polarization direction makes an angle α with the optical LC axis, then the total transmission coefficient Tinter for the electromagnetic wave is calculated using the expression (3). When the unpolarized electromagnetic radiation falls on the LC cell, the energy transmission coefficient Tinter is calculated using the formula (4). At the fourth stage, the transmission coefficient of the entire LC modulator of electromagnetic radiation TM is calculated using the expressions (1).
On the basis of the described algorithm, the programm lc the Fortran 95 language was developed for calculation of energy transmission coefficient of the LCD modulator operating on the basis of the typical π-cell [10]. The input parameters of the lc program are the following parameters of the LC cell: refractive indices of the medium (n1, n9) where the cell is located; refractive index and thickness of the glass substrates (n2, n8, l2, l8); refractive index and thickness of the electrode layers (n3, n7, l3, l7); refractive index and thickness of the orienting layers (n4, n6, l4, l6); refractive indices of the LC (no, ne) and thickness of its layer llc. In addition, the input program parameter is the wavelength of the modulated electromagnetic radiation λ and the initial distribution of orientation angles of the LC director θ(x), as well as the control modulator voltage. The output parameter of this program is the transmission coefficient value of the entire LC modulator TM.
Results and discussion
To test the proposed calculation method for the transmission coefficient of the LC modulator, a comparison was made of the simulation results related to the device specifications obtained using the described method with the results obtained using the well-known MOUSE – LCD software package [13]. The physical parameters of the LC mixture and the process parameters of the LC cell that have been used in the calculations, are given in [9, 14]. The calculations were performed for a radiation with a wavelength λ = 0.5 μm.
Fig. 2 shows the simulation results of the dependence of the LC modulator transmission coefficient TM on the control voltage U, obtained using the proposed method and the MOUSE-LCD software package. The discrepancy between these data does not exceed 10% that corresponds to the calculation accuracy of the MOUSE-LCD software package and indicates the adequacy of the proposed simple method for calculating the transmission coefficient of the LC modulator based on the π-cell.
One of the main tasks in designing the LC modulators is to minimize the losses related to the available Fresnel losses due to reflection from the plane-parallel layers with various refractive indices [15]. In this case, as a rule, the dependence of these losses is considered only in relation to the design parameters of the LC cell (see, for example, [10]), and the influence of the control voltage is not taken into account. However, such losses depend on the nature of the multi-wave interference in the layered LC cell structure, and, therefore, also depend on the extraordinary refractive LC index which value is changed as the control voltage evolves. Therefore, the issue of influence of the control LC cell voltage on the losses related to the multipath interference in the layered cell structure is of great interest. Figure 3 shows the dependences of the transmission coefficient of an LC cell without any polarizers on the control voltage. It is clear that the Fresnel reflection losses with due regard to the multi-wave interference, depends quite strongly on the control voltage and the modulated radiation wavelength. The difference between the loss values, for example, for the wavelengths of 0.5 μm and 16 μm can reach 50%. The dependence nature is also determined by the wavelength of electromagnetic radiation passing through the LC cell. Thus, for the radiation wavelengths of 0.5, 1, 2 and 8 μm, the transmission coefficient Tinter with an increased control voltage U has the shape of a curve with a minimum, and for the wavelengths of 4 and 16 μm, the dependence Tinter(U) has the shape of a curve with a maximum. The value of changes can be about 15–20% of the maximum value while indicating the possible modulation of electromagnetic radiation even in a polaroid-free modulator structure.
However, in this case, the amplitude modulation coefficient m does not exceed 11%. Two conclusions can be drawn:
the value of Fresnel losses in the LC cell can be adjusted by selecting the control voltage in the condition with maximum transmission;
if the modulation of electromagnetic radiation is required within a wide range of wavelengths, but with a low amplitude modulation coefficient, then only one non-polaroid LC cell can be used for this purpose.
To obtain higher values of the amplitude modulation coefficient, it is necessary to use a modulator design with polaroids. In this case, for THz radiation, the optimal thickness of the working clearance of the LC cell llc is determined based on the maximum interference of polarized waves [4]:
λ0 = 2 · ne − no · llc, (5)
where λ0 is the modulated radiation wavelength.
In this case, the standard structure of the THz radiation LC modulator based on only one π-cell has one significant drawback: it has too long full response time [4]. Previously, a more comprehensive design of the LC modulator based on N identical elementary π-cells was proposed to solve this issue. Each of these elementary cells has its own electric control, and the entire set of such cells is placed between two polarizers. The number of such π-cells N is determined by the condition of maximum interference of the polarized waves [4]:
λ0 = 2 · N · ne − no · l0, (6)
where L0 is the thickness of one elementary π-cell.
The table demonstrates the results of computer simulation of the specifications of THz-radiation LC modulators for various wavelengths. It should be noted that each LC modulator is designed only for a certain wavelength and has a different number of elementary LC cells. The following set of specifications was selected to describe the LC modulator, such as an amplitude modulation coefficient m, maximum open-state transmission coefficient Tmax and total response time τ. The modulators differ in the number N of elementary π-cells. It follows from the table that the LC modulators for various wavelengths may differ insignificantly only in the maximum transmittance value, with all other specifications being equal. This is explained by the fact that all modulators are based on the same elementary LC cell and satisfy the polarized wave interference condition (5). The difference in the maximum transmittance values is resulted from the losses in multilayer structures with the increased interfaces.
Conclusion
This paper proposes a simple matrix calculation method for the energy transmission coefficient of the LC modulator based on a typical π-cell. The method is based on the matrix formalism of comprehensive 2 × 2 Abeles matrices and allows for simultaneous consideration of the multipath interference in the layered cell structure and the phenomenon of electric field-controlled birefringence in the LC. Comparison of the data obtained using the method described above with the calculation results obtained by the proven software packages has shown their good quantitative fit. This allows us to confirm the adequacy and practical suitability of the proposed method to simulate the specifications of the LC modulator based on a π-cell. The simulation results have showed that the losses in the LC cell that are caused by the Fresnel reflections from the cell structure interfaces and multi-wave interference in this structure, depends on the control voltage and is different for various wavelengths of the electromagnetic range. The difference between the loss value, for example, for the wavelengths of 0.5 μm and 16 μm, can reach 50%.
In addition, dependence of the energy transmission coefficient of the LC cell on the control electric field can be used to modulate the electromagnetic radiation of various ranges with a small value of the amplitude modulation coefficient. It has been shown by the computer method that the application of the LC modulator structure consisting of a certain number of identical π-cells allows achieving acceptable parameters of the modulated THz radiation. In this case, the number of elementary LC cells is determined by the maximum interference of polarized waves, wavelength of the modulated THz radiation and thickness of the elementary LC cell.
AUTHOR
Simonenko Georgy V., Dr. of Sc. (Phys.&Math.), Prof. Department of Optics and Biophotonics, Chernyshevsky Saratov National Research State University, e-mail: simonenkogv@sgu.ru; Saratov, Russia.
ORCID: 0000-0002-6283-6335
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