Open AccessArticle

Research on the Modulation Characteristics of LiNbO₃ Crystals Based on the Three-Dimensional Ray Tracing Method

Houpeng Sun

¹,

Yingchun Li

^2,*,

Laixian Zhang

^2,*,

Huichao Guo

²,

Chenglong Luan

¹,

Haijing Zheng

²,

Rong Li

² and

Youchen Fan

Graduate School, Space Engineering University, Beijing 101416, China

Department of Electronic and Optical Engineering, Space Engineering University, Beijing 101416, China

School of Space Information, Space Engineering University, Beijing 101416, China

Authors to whom correspondence should be addressed.

Crystals 2024, 14(12), 1101; https://doi.org/10.3390/cryst14121101 (registering DOI)

Submission received: 18 November 2024 / Revised: 8 December 2024 / Accepted: 19 December 2024 / Published: 20 December 2024

(This article belongs to the Section Inorganic Crystalline Materials)

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Abstract

To further study the electro-optical modulation characteristics of LiNbO₃ crystals and analyze their modulation performance, a method for studying the modulation characteristics of LiNbO₃ crystals, based on the three-dimensional ray tracing method, is introduced. With the help of the refractive index ellipsoidal theory, the optical properties of LiNbO₃ crystals under the influence of the Pockels effect are systematically investigated. The research results show that the optical properties of LiNbO₃ crystals under the action of an external electric field can be divided into two cases: the crystal optical axis is parallel to the clear light direction, and the crystal optical axis is perpendicular to the clear light direction. Subsequently, starting from Maxwell’s equation and the matter equation, the analytical expressions of optical parameters such as refractive index, wave vector, light vector, optical path, and phase delay in electro-optical crystals are derived. Finally, the propagation law of LiNbO₃ crystals when the light is incident in any direction, i.e., when the optical axis of the crystal is parallel to the clear direction and perpendicular to the clear direction, and the light intensity and field of view of the LiNbO₃ crystal for electro-optical modulation are discussed.

Keywords:

modulation characteristics; LiNbO₃ crystal; three-dimensional ray tracing

1. Introduction

Crystals are important basic materials in modern science, and according to their functional effects, crystals can be divided into laser crystals, nonlinear optical crystals, piezoelectric crystals, pyroelectric crystals, electro-optical crystals, magneto-optical crystals, etc. Electro-optical crystals can change their optical properties by applying an external electric field and have a wide range of application prospects in many fields, such as optical communication, optical storage, laser processing, and laser modulation. Under an applied electric field, the crystal changes from isotropic to anisotropic, resulting in birefringence called the electro-optical effect. The crystal’s electro-optic effect changes the crystal’s light wave characteristics, enabling amplitude and phase modulation of the incident rays. Crystals with electro-optical effects are called electro-optical crystals.

The ray tracing method can describe the propagation of light within the crystal. Jones matrices [1,2,3,4,5,6] and Muller matrices [7,8,9,10,11,12,13] commonly describe the polarization state changes of polarized light propagating in a two-dimensional coordinate system. However, these two methods are unsuitable for 3D imaging scenarios with incident angles. Yun, G., used the orthogonal basis vector to convert the Jones vector in the two-dimensional coordinate system to the three-dimensional coordinate system and established a three-dimensional ray tracing model of light propagation in an isotropic medium [14,15]. Zhang [16] and Y. Li et al. [17] successfully proposed solving the polarization state of incident light propagating in an anisotropic medium using a three-dimensional coherence matrix. However, the electro-optical properties of biaxial crystals were not analyzed. In 2018, Professor Russell A. Chipman of the University of Arizona published a monograph on polarized light propagation [18]. In 2020, Dongsheng Song of Shanghai Jiao Tong University proposed an intrinsic generalized Jones matrix method to describe the polarization state and phase change of fully polarized light propagating in nonlinear crystals [19]. In this method, the idea of differentiation is introduced into the derivation of generalized Jones matrices of nonlinear crystals, but differential generalized Jones matrices cannot calculate the polarization change of oblique incident light or manage the scenario in which the optical axis does not coincide with the laboratory coordinates. In 2023, Li Yahong of the Dalian University of Technology proposed a three-dimensional polarized ray tracing Muller algorithm for any surface-type optical system [17,20]. In this method, the ZEMAX Standard 2016 optical software is used to trace the three-dimensional polarized light of each optical interface, and the three-dimensional Muller matrix of each optical interface is calculated in its respective local coordinate system. Dongyang Jiang described a novel synthesis method for LiNbO₃ crystals, which improved the growth quality and efficiency of the crystals [21]. Yiyang Wen described the fabrication and photonic application of LiNbO₃ crystal thin films for the surface coating of optoelectronic devices [22]. In 2024, Wei Peng provided an introduction into the piezoelectric properties and thermal stability of the LiNbO₃ crystals [23]. Elmaataouy, E., uses LiNbO₃ as a coating to improve the cycling stability and high rate capability of the lithium-ion battery anode material [24].

There are three commonly used crystals, i.e., KDP, BBO, and LiNbO₃ crystals. KDP crystals belong to the tetragonal crystal system, and their light transmission band is 178 nm–1.45 μm, the nonlinear optical coefficient d₃₆ (1.064 μm) = 0.39 pm/V, the birefringence coefficient is high, the electro-optical coefficient is high, the Mohs hardness is 2.5, the hardness is low, and the polished surface is easy to dissolve; they are mainly used in the field of laser technology and high-power laser systems. BBO crystals belong to a trigonal crystal system, with a light transmission range of 190 nm–3500 nm, a nonlinear coefficient of about four times that of KDP d₃₆, moderate birefringence characteristics, a significant electro-optical effect, Mohs hardness of 4.5–5, and good mechanical and chemical stability; they are used in laser technology, quantum computing, communications, medical imaging, and other fields. The LiNbO₃ crystals have a perovskite structure and cubic crystal symmetry, although their light transmission range is relatively narrow, the nonlinear optical coefficient is small, the birefringence characteristics are weak, and the chemical stability is good, they change easily in humid air, and the mechanical properties are brittle, but they display metal conductivity and high electrical conductivity. In terms of electro-optical modulation, the advantage of LiNbO₃ crystals lies in their unique electrical properties, which can provide modulation methods that are different from those of traditional electro-optical crystals. Their high conductivity makes it possible to use their electrical properties to interact with light in electro-optical modulation, and it is possible to achieve some special modulation effects. In addition, although their nonlinear optical coefficients and birefringence characteristics are weak, they can be compensated for by compounding them with other materials or adopting a special structural design, thus bringing new development opportunities to the field of electro-optical modulation.

In this paper, according to the refractive index ellipsoid theory and the Fresnel equation, a three-dimensional ray tracing model of polarized light propagation in LiNbO₃ crystal under arbitrary incidence conditions is constructed, and the Pockels effect of LiNbO₃ crystals is analyzed. By constructing a spatial three-dimensional coordinate system, the birefringence effect of LiNbO₃ crystals was studied using the ray tracing method. Compared with previous studies, the study in this paper can comprehensively and accurately analyze the electro-optical effect of LiNbO₃ crystals under arbitrary conditions. In addition, the electro-optical modulation characteristics of LiNbO₃ crystals are systematically analyzed, which provides a theoretical basis for the subsequent research of electro-optical crystals.

2. Electro-Optical Effect of LiNbO₃ Crystals

The nonlinear magnetic susceptibility of LiNbO₃ crystals is generally reflected by nonlinear optical coefficients, such as 5%MgO-doped LiNbO₃ at 1064 nm, d₃₁ = 4.4 pm/V, d₃₃ = 25 pm/V. The anisotropy of the crystal structure leads to different nonlinear optical coefficients, e.g., d₃₃ corresponds to the interaction parallel to the z-axis, and a larger value indicates that the nonlinear optical effect in this direction is more significant, which is related to the ion arrangement and bonding mode in the crystal structure. Doping different elements can change their optical properties and nonlinear optical coefficients, e.g., MgO doping improves optical and photorefractive resistance and affects nonlinear optical coefficients, making them more suitable for high-power applications. In applications such as laser technology, the nonlinear optical coefficients of LiNbO₃ crystals play a key role, promoting efficient frequency doubling conversion.

2.1. Refractive Index Ellipsoid Analytical Model

There are many methods to study the electro-optical effect of crystals, and the refractive index ellipsoidal method can vividly describe the spatial distribution of the refractive index before and after the electric field is applied to the crystal, which is widely used in the analysis of the modulation characteristics of electro-optical crystals. In the absence of an external electric field, the refractive index ellipsoidal equation of a crystal can be expressed as follows:

\frac{X_{0}^{2}}{n_{X 0}^{2}} + \frac{Y_{0}^{2}}{n_{Y 0}^{2}} + \frac{Z_{0}^{2}}{n_{Z 0}^{2}} = 1,

(1)

where X₀, Y₀, and Z₀ represent the dielectric spindle of the crystal when there is no external electric field;

n_{X 0}

n_{Y 0}

, and

n_{Z 0}

are the principal refractive indices in the X₀, Y₀, and Z₀ directions, respectively. The refractive index ellipsoid, which describes the refractive index of a crystal, is shown in Figure 1 below.

According to the refractive index ellipsoid, the crystal’s refractive index in all directions can be calculated intuitively and visually. Using the refractive index of the three main axes of the refractive index ellipsoid, the crystal family can be determined, and the method is shown in Table 1 below.

The crystals of the higher crystal family exhibit complete symmetry and multiple higher axes, and the light propagates on the crystal surface according to Snell’s law; a beam of incident light is refracted through the crystal surface to produce a refracted light. The crystals of the intermediate crystal family are anisotropic, contain a higher order axis, have a single optical axis, and are uniaxial crystals. When the incident light propagates at the surface of the crystal, a birefringence effect occurs, producing two refracted rays, i.e., one ray of light that conforms to Snell’s law is called ordinary O-light, and the other ray that does not satisfy Snell’s law is called abnormal E-light. The crystals of the lower crystal group are also anisotropic; there is no higher order axis, and there are two crystal optical axes, called biaxial crystals. When the incident light propagates at the surface of the crystal, a birefringence effect occurs, producing two abnormal beams of light, i.e., fast light, and slow light.

When an electric field is applied at both ends of the electro-optical crystal, the refractive index in the crystal changes due to the electro-optical effect, and the generalized refractive index ellipsoid of the electro-optical crystal can be expressed as follows:

\begin{array}{l} (\frac{1}{n_{X 0}^{2}} + Δ β_{1}) X_{0}^{2} + (\frac{1}{n_{Y 0}^{2}} + Δ β_{2}) Y_{0}^{2} + (\frac{1}{n_{Z 0}^{2}} + Δ β_{3}) Z_{0}^{2} \\ + 2 Δ β_{4} Y_{0} Z_{0} + 2 Δ β_{5} X_{0} Z_{0} + 2 Δ β_{6} X_{0} Y_{0} = 1 . \end{array}

(2)

According to Equation (2), the coefficients of the major axes of the crystal X₀, Y₀, and Z₀ axes in the original refractive index are changed, so the refractive index of the three main axes also changes. In addition, due to the presence of the cross terms Y₀Z₀, X₀Z₀, and X₀Y₀, in Equation (2), the refractive index major axis of the crystal will be deflected.

Δ β_{i} (i = 1, 2, \dots, 6)

in Equation (2) is the amount of dielectric tensor change in the crystal caused by the electro-optical effect of the crystal, composed of two parts, which can be expressed as follows:

Δ β = [\begin{array}{l} Δ β_{1} \\ Δ β_{2} \\ Δ β_{3} \\ Δ β_{4} \\ Δ β_{5} \\ Δ β_{6} \end{array}] = γ_{pockels} [\begin{array}{l} E_{X 0} \\ E_{Y 0} \\ E_{Z 0} \end{array}] + h_{kerr} [\begin{array}{l} E_{X 0}^{2} \\ E_{Y 0}^{2} \\ E_{Z 0}^{2} \\ E_{Y 0} E_{Z 0} \\ E_{X 0} E_{Z 0} \\ E_{X 0} E_{Y 0} \end{array}] .

(3)

Equation (3) is divided into two parts: the first part is the dielectric tensor change caused by the Pockels effect of the crystal, in which γ_pockels is the linear electro-optical coefficient (or Pockels coefficient) matrix, and E_X₀, E_Y₀, and E_Z₀ are the components of the applied electric field in the direction of the three principal dielectric axes of the crystal, respectively. The second half of Equation (3) is the amount of dielectric tensor change caused by the crystal Kerr effect, in which h_kerr is the coefficient matrix that characterizes the crystal Kerr effect, called the secondary electro-optical coefficient (or Kerr coefficient) matrix. When analyzing the electro-optical effect of a crystal, the Kerr effect and Pockels effect of the crystal can be studied independently, and the influence of the electro-optical effect on the performance of the crystal can be comprehensively analyzed.

In practice, the dielectric tensor of the crystal caused by the applied electric field does not change much. However, it can change the symmetry of the crystal refractive index ellipsoid, modulating the internal light field of the crystal by controlling the applied external field. Various optical modulation devices can be fabricated by using electro-optical crystals, such as high-speed electro-optical open light, electro-optical scanners, electro-optical polarizers, and electro-optical modulators.

The following factors are mainly considered when selecting a modulated electro-optical crystal: (1) The crystal must exhibit good growth, i.e., obtaining high-quality large-size crystals. In electro-optic modulation, it is necessary to control the on-off of the modulated light source through the crystal. Hence, the crystal needs a large clear aperture to ensure the integrity of the light source information and prevent leakage. (2) The crystal must show good stability performance, i.e., the crystal needs good thermal conductivity, which helps it dissipate heat and avoid damage to its performance due to overheating. The crystal needs to be able to withstand high voltage to prevent it from being broken down during the modulation process. (3) The crystal must possess a large electro-optical coefficient, and the crystal selected in the project is used to achieve the electro-optical modulation switch, in which half-wave voltage of the crystal with large electro-optical coefficient is lower. (4) The crystal must exhibit a large refractive index, good uniformity, and high transmittance to ensure the imaging performance of the crystal. (5) The crystal must exhibit stable physical and chemical properties under different environmental conditions, e.g., the crystal can maintain a stable structure and performance regardless of temperature, humidity, chemical corrosion, etc. The most suitable crystal, with comprehensive properties, must be chosen according to the application requirements.

From the practical application perspective, KDP (KH₂PO₄) and LN (LiNbO₃) crystals display superior comprehensive properties and are the most widely used. KDP crystals exhibit a large electro-optical coefficient, easy growth, a high damage threshold, and a spectral band from 1550 nm to 2150 nm, and they are widely used in high-power lasers. However, because KDP crystals are easily dissolved, it is necessary to pay attention to environmental conditions when using them, and they need to be sealed and protected when compared. Compared with KDP crystals, LiNbO₃ crystals exhibit good mechanical properties, are not easy to dissolve, and display a larger electro-optical coefficient and refractive index under the same wavelength conditions, and they are also commonly used electro-optical modulation crystals. In addition, KTN (KTa_1-xNb_xO₃, potassium tantalum niobate) crystals exhibit a significant secondary electro-optical effect, a large electro-optical coefficient, and a low half-wave voltage, but it is not easy to generate large-size high-uniformity crystals, so the wide application of these crystals needs to be further explored.

LiNbO₃ crystals exhibit excellent performance and are more suitable for laser 3D imaging applications, so LiNbO₃ was selected for electro-optical modulation research. The performance parameters of LiNbO₃ crystals are shown in Table 2 below.

2.2. The Pockels Effect of LiNbO₃ Crystals

In order to study the electro-optical modulation characteristics of LiNbO₃ crystals, it is necessary to analyze the electro-optical effect of the crystals using the refractive index ellipsoidal method. The LiNbO₃ crystal is a uniaxial crystal without adding voltage, and the optical axis of the crystal is along the Z₀ axis. The refractive index ellipsoidal equation for a LiNbO₃ crystal without added voltage can be expressed as follows:

\frac{X_{0}^{2}}{n_{o}^{2}} + \frac{Y_{0}^{2}}{n_{o}^{2}} + \frac{Z_{0}^{2}}{n_{e}^{2}} = 1 .

(4)

Among them, n_o and n_e are the refractive indices of ordinary light and abnormal light of the LiNbO₃ crystals, respectively; the refractive indices of the crystal in the two directions of the refractive index principal axis X₀ and Y₀ are equal to n_o, and the refractive index in the direction of the refractive index principal axis Z₀ is n_e.

When a voltage is applied at both ends of the LiNbO₃ crystal, the crystal has a pronounced Pockels effect. Bringing the Pockels coefficient matrix of LiNbO₃ crystals into the dielectric isolation equation yields the amount of change of the applied voltage to the dielectric isolation tensor of the crystal, as follows:

Δ β = [\begin{array}{l} Δ β_{1} \\ Δ β_{2} \\ Δ β_{3} \\ Δ β_{4} \\ Δ β_{5} \\ Δ β_{6} \end{array}] = γ_{pockels} [\begin{array}{l} E_{X 0} \\ E_{Y 0} \\ E_{Z 0} \end{array}] = [\begin{matrix} 0 & - γ_{22} & γ_{13} \\ 0 & γ_{22} & γ_{13} \\ 0 & 0 & γ_{33} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ - γ_{22} & 0 & 0 \end{matrix}] [\begin{array}{l} E_{X 0} \\ E_{Y 0} \\ E_{Z 0} \end{array}] = [\begin{matrix} - γ_{22} E_{Y 0} + γ_{13} E_{Z 0} \\ γ_{22} E_{Y 0} + γ_{13} E_{Z 0} \\ γ_{33} E_{Z 0} \\ 0 \\ 0 \\ - γ_{22} E_{X 0} \end{matrix}] .

(5)

Among them, γ₁₃, γ₂₂, and γ₃₃ are the electro-optical coefficients of the LiNbO₃ crystal, and E_X₀, E_Y₀, and E_Z₀ are the electric field strengths applied by the LiNbO₃ crystal in the direction of X₀, Y₀, and Z₀, respectively. Bringing Equation (5) into the general expression of the refractive index ellipsoid, we can obtain the refractive index ellipsoid equation for LiNbO₃ crystals after an electric field is applied, as follows:

\begin{array}{l} (\frac{1}{n_{o}^{2}} - γ_{22} E_{Y 0} + γ_{13} E_{Z 0}) X_{0}^{2} + (\frac{1}{n_{o}^{2}} + γ_{22} E_{Y 0} + γ_{13} E_{Z 0}) Y_{0}^{2} + (\frac{1}{n_{e}^{2}} + γ_{33} E_{Z 0}) Z_{0}^{2} \\ - 2 γ_{22} E_{X 0} X_{0} Y_{0} = 1 . \end{array}

(6)

In Equation (6), in addition to having refractive index components along the three directions of the refractive index principal axes X₀, Y₀, and Z₀, there are also some crossover terms. Therefore, it is necessary to normalize the refractive index ellipsoid equation to determine the direction of the principal axis of the refractive index after the electric field is applied to the crystal and to calculate the refractive index in the corresponding direction.

In the following section, according to the direction of the applied voltage during the electro-optical modulation of the LiNbO₃ crystal, the corresponding electro-optical effect is analyzed using the refractive index ellipsoidal method.

(1): The LiNbO₃ crystal is subjected to a voltage along the X₀ axis.

When a voltage is applied to a LiNbO₃ crystal along the X₀ axis, E_Y₀ = E_Z₀ = 0, E_X₀ ≠ 0, the general form of the refractive index ellipsoid equation can be expressed as follows:

\frac{X_{0}^{2}}{n_{o}^{2}} + \frac{Y_{0}^{2}}{n_{o}^{2}} + \frac{Z_{0}^{2}}{n_{e}^{2}} - 2 γ_{22} E_{X 0} X_{0} Y_{0} = 1 .

(7)

In order to analyze the principal axis direction of the refractive index ellipsoid of the energized crystal, it is necessary to perform the spindle operation on Equation (7) to eliminate the two cross terms in the equation. Firstly, the dielectric spindle of the LiNbO₃ crystal is rotated θ along the Z₀ axis, and the X₀Y₀ cross term is eliminated; the new spindle is X₀′Y₀′Z₀′. The relationship between X₀Y₀Z₀ and X₀′Y₀′Z₀ can be expressed as follows:

{\begin{cases} X_{0} = {X_{0}}^{'} \cos θ - {Y_{0}}^{'} \sin θ \\ Y_{0} = {X_{0}}^{'} \sin θ + {Y_{0}}^{'} \cos θ \\ Z_{0} = {Z_{0}}^{'} \end{cases} .

(8)

Bringing Equation (8) into Equation (7) yields the following:

\begin{array}{l} (\frac{1}{n_{o}^{2}} - γ_{22} E_{X 0} \sin 2 θ) {X_{0}}^{'}^{2} + (\frac{1}{n_{o}^{2}} + γ_{22} E_{X 0} \sin 2 θ) {Y_{0}}^{'}^{2} + \frac{{Z_{0}}^{'}^{2}}{n_{e}^{2}} \\ + 2 γ_{22} E_{X 0} {X_{0}}^{'} {Y_{0}}^{'} (\sin^{2} θ - \cos^{2} θ) = 1 . \end{array}

(9)

Set

\sin^{2} θ - \cos^{2} θ = 0

in Equation (9) to eliminate the intersection term X₀′Y₀′ and obtain θ = ±45°. It should be pointed out that θ has nothing to do with the strength of the voltage applied to both ends of the crystal; usually, take θ = 45°, and then Equation (9) can be simplified as follows:

(\frac{1}{n_{o}^{2}} - γ_{22} E_{X 0}) x_{1}^{2} + (\frac{1}{n_{o}^{2}} + γ_{22} E_{X 0}) x_{2}^{2} + \frac{x_{3}^{2}}{n_{e}^{2}} = 1 .

(10)

In Equation (10), x₁x₂x₃ is obtained by rotating the refractive index principal axis X₀Y₀Z₀ around the Z₀ axis = 45° when the LiNbO₃ crystal is not charged.

The electro-optical coefficient γ₂₂ of the LiNbO₃ crystal is of the order of magnitude 10⁻¹² m/V, and the maximum value of the electric field E_X₀ applied at both ends of the crystal is usually of the order of magnitude 10⁶ V/m, so

γ_{22} E_{X 0} ≪ 1

. According to the Taylor function formula, γ₂₂E_X₀ is expanded, and the first two terms are taken; Equation (10) can be expressed as follows:

\frac{x_{1}^{2}}{{(n_{o} + \frac{1}{2} n_{o}^{3} γ_{22} E_{X 0})}^{2}} + \frac{x_{2}^{2}}{{(n_{o} - \frac{1}{2} n_{o}^{3} γ_{22} E_{X 0})}^{2}} + \frac{x_{3}^{2}}{n_{e}^{2}} = 1 .

(11)

The amount of change in the refractive index caused by the electric field applied at both ends of the crystal can be expressed as

n_{E} = n_{o}^{3} γ_{22} E_{X 0} / 2

. When the voltage is applied to the LiNbO₃ crystal along the X₀ axis, the refractive indices of the three refractive index axes are different, and the refractive index changes from uniaxial crystals to biaxial crystals.

When the incident light is incident perpendicular to the x₃ axis, the refracted light wave is only related to the voltage applied at both ends of the crystal. It is not affected by the crystal birefringence effect, so when the voltage is applied along the X₀ axis, the crystal direction is generally set to be parallel to the x₃ axis. Since the x₃ axis direction is the same as the Z₀ axis, the clear direction of the crystal is parallel to the Z₀ axis, that is, parallel to the axis direction when the crystal is not energized.

(2): The LiNbO₃ crystal is subjected to a voltage along the Y₀ axis.

When the LiNbO₃ crystal is energized in the direction of the Y₀ axis, E_X₀ = E_Z₀ = 0, E_Y₀ ≠ 0, the general form of the refractive index ellipsoid equation can be expressed as follows:

(\frac{1}{n_{o}^{2}} - γ_{22} E_{Y 0}) X_{0}^{2} + (\frac{1}{n_{o}^{2}} + γ_{22} E_{Y 0}) Y_{0}^{2} + (\frac{1}{n_{e}^{2}}) Z_{0}^{2} = 1 .

(12)

According to the Taylor function formula,

γ_{22} E_{X 0}

is expanded, the first two terms are taken, and Equation (12) can be expressed as follows:

\frac{x_{1}^{2}}{{(n_{o} + \frac{1}{2} n_{o}^{3} γ_{22} E_{Y 0})}^{2}} + \frac{x_{2}^{2}}{{(n_{o} - \frac{1}{2} n_{o}^{3} γ_{22} E_{Y 0})}^{2}} + \frac{x_{3}^{2}}{n_{e}^{2}} = 1 .

(13)

The amount of refractive index change caused by the application of an electric field at both ends of a LiNbO₃ crystal can be expressed as follows:

n_{E} = n_{o}^{3} γ_{22} E_{X 0} / 2

. The crystal changes from a uniaxial crystal to a biaxial crystal, and the direction of light is usually set parallel to the x₃ axis, parallel to the optical axis of the crystal without an external electric field.

When the crystal is energized along the Y₀ axis, the induction spindle x₁x₂x₃ coincides with the crystal dielectric spindle X₀Y₀Z₀ when the unenergized crystal is powered, and the direction of the new refractive index spindle x₁x₂x₃ does not change.

(3): The LiNbO₃ crystal is subjected to a voltage along the Z₀ axis.

When a voltage is applied to a LiNbO₃ crystal along the Z₀ axis, E_X₀ = E_Y₀ = 0, E_Z₀ ≠ 0, the general form of the refractive index ellipsoidal equation of the crystal can be expressed as follows:

(\frac{1}{n_{o}^{2}} + γ_{13} E_{Z 0}) X_{0}^{2} + (\frac{1}{n_{o}^{2}} + γ_{13} E_{Z 0}) Y_{0}^{2} + (\frac{1}{n_{e}^{2}} + γ_{33} E_{Z 0}) Z_{0}^{2} = 1 .

(14)

Equation (14) does not have a cross term, which means that the principal refractive index of the crystal does not rotate, but the refractive index of the principal shaft changes. The electro-optical coefficients γ₁₃ and γ₃₃ are of the order of magnitude 10⁻¹² m/V, and E_Z₀ is of the order of magnitude 10⁶ V/m, so

γ_{13} E_{Z 0} ≪ 1

and

γ_{33} E_{Z 0} ≪ 1

; according to the Taylor function formula, γ₁₃E_Z₀ and γ₃₃E_Z₀ are expanded, the first two terms are taken, and Equation (14) can be expressed as follows:

\frac{x_{1}^{2}}{{(n_{o} - \frac{1}{2} n_{o}^{3} γ_{13} E_{Z 0})}^{2}} + \frac{x_{2}^{2}}{{(n_{o} - \frac{1}{2} n_{o}^{3} γ_{13} E_{Z 0})}^{2}} + \frac{x_{3}^{2}}{{(n_{e} - \frac{1}{2} n_{e}^{3} γ_{33} E_{Z 0})}^{2}} = 1 .

(15)

The amount of change in the refractive index due to the application of voltage across the LiNbO₃ crystal can be expressed as follows:

n_{E 1} = n_{o}^{3} γ_{13} E_{Z 0} / 2

n_{E 2} = {(n_{o} - n_{Δ})}^{3} γ_{33} E_{Z 0} / 2

. When the LiNbO₃ crystal is energized along the Z₀ axis, the crystal spindle does not rotate, but the refractive index changes in the direction of the spindle.

Based on the analysis of the optical properties of the LiNbO₃ crystal applying voltage in all directions, the Pockels effect of the LiNbO₃ crystal is summarized in Table 3 below.

In summary, the use of LiNiO₃ crystals for electro-optical modulation comprises the following situations. When the direction of the light is along the Z-axis (the direction of the axis when the crystal is not energized), if the direction of the external electric field is also along the optical axis, the two refracted rays corresponding to the perpendicular incident rays have the same refractive index. There is no phase difference in the propagation process of the crystal, so it is not easy to achieve an effective modulation effect. Therefore, when the direction of light is along the Z-axis, the direction of the electric field should be either the axis or the axis direction, and the modulation effect of the two methods is the same. When an electric field is applied along the axis, the refractive index ellipsoidal coordinate system will rotate 45° around the Z axis to become an O-X′Y′Z′ coordinate system, and this electro-optical modulation method is called the “light along the optical axis”. In the same way, when the direction of the electric field is along the Z-axis, the direction of the light should be the X-axis or Y-axis, and the modulation effect of the two methods is the same. In this paper, we take the direction of light transmission along the X-axis as an example and call this electro-optical modulation method the “perpendicular optical axis direction”.

LiNbO₃ belongs to the C_3v–3m crystal class. In the practical application process, the Kerr effect of LiNbO₃ crystals is relatively weak compared with the Pockels effect, so the Kerr effect of the crystal will not be elaborated in this paper.

3. Three-Dimensional Ray Tracing Method for LiNbO₃ Crystal Modulation

The traditional ray tracing method only analyzes the propagation characteristics of the rays in the two-dimensional plane, ignoring the changes in the characteristics of the rays in the vertical direction. In the actual electro-optical crystal, the propagation of light is three-dimensional, and the three-dimensional light can accurately simulate the propagation path, refraction, and reflection of light in the crystal and comprehensively consider the changes of light in all directions so as to accurately reflect the modulation effect of the crystal on the light. The modulation effect of electro-optical crystals on light is not only reflected in the change of the refractive index, but it also affects the polarization state of light. The three-dimensional ray tracing method can analyze the polarization state changes of light rays in crystals in detail, which is important for understanding the modulation mechanism of electro-optical crystals. In addition, tracing the propagation of light rays through the crystal makes it possible to accurately calculate the phase change of light as it passes through the crystal. Phase information is essential for understanding the interference and diffraction of light, as well as the performance of optical systems, helping to optimize the design and performance of electro-optical modulation devices. Therefore, based on the refractive index ellipsoid, the electro-optical modulation characteristics of LiNbO₃ crystals are analyzed using the three-dimensional ray tracing method.

3.1. The 3D Coordinate System Construction

When using the three-dimensional ray tracing method to study the modulation performance of electro-optical crystals, it is necessary to construct the three-dimensional coordinate system of the crystals. The three-dimensional coordinate system of the crystal can completely capture the propagation behavior of light in all directions in the crystal, accurately describe the propagation path of light, and accurately define the polarization direction vector change of light.

In Section 2, we take a closer look at the electro-optical effects of LiNbO₃ crystals. After careful analysis, the following conclusions are drawn: LiNbO₃ crystals have two modulation methods. First, the light is transmitted along the optical axis of the crystal in the uncharged state; second, the light is transmitted perpendicular to the direction of the crystal when it is not energized. In this section, we will use the three-dimensional ray tracing method to physically simulate the modulation process of the two modulation methods of LiNbO₃ crystals.

When the modulation mode of the LiNbO₃ crystal assumes that “the optical axis of the crystal is parallel to the direction of the light”, after the voltage is applied at both ends of the crystal, it changes from a uniaxial crystal to a biaxial crystal. The optical axis of the crystal rotates 45 degrees around the Z-axis, and the three-dimensional coordinate system of the crystal is shown in Figure 2 below.

In Figure 2, O-xyz is the geometric coordinate system of the crystal, which is established by the crystal’s geometry. The x-axis, y-axis, and z-axis are along the crystal’s geometric edge and are perpendicular to each other.

In the geometric coordinate system O-xyz of LiNbO₃ crystals, an incident ray k_i enters the crystal from the air, and the angle between the negative direction of the ray in the z-axis is called the angle of incidence α, and the angle between the ray in the positive direction of the x-axis is called the azimuth ϕ. Thus, the incident ray k_i can be expressed as follows:

k_{i} = (\begin{matrix} \hat{x} & \hat{y} & \hat{z} \end{matrix}) [\begin{matrix} \sin α \cos ϕ \\ \sin α \sin ϕ \\ \cos α \end{matrix}],

(16)

where

\hat{x}

\hat{y}

, and

\hat{z}

are the unit direction vectors of the three axes in the geometric coordinate system O-xyz of LiNbO₃ crystals, respectively.

When no voltage is applied at both ends of the LiNbO₃ crystal, the optical axis of the crystal is parallel to the z-axis, and the dielectric principal axis coordinate system O-X₀Y₀Z₀ coincides with the crystal geometric coordinate system O-xyz. After the modulation voltage is applied at both ends of the LiNbO₃ crystal, the dielectric spindle coordinate system O-X₀Y₀Z₀ rotates θ = 45° around Z₀ to obtain the induced spindle coordinate system O-x₁x₂x₃, and the relationship between the induced spindle coordinate system x₁x₂x₃ and the crystal geometric coordinate system xyz can be expressed as follows:

(\begin{matrix} {\hat{x}}_{1} & {\hat{x}}_{2} & {\hat{x}}_{3} \end{matrix}) = (\begin{matrix} \hat{x} & \hat{y} & \hat{z} \end{matrix}) [\begin{matrix} \cos θ & - \sin θ & 0 \\ \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{matrix}] .

(17)

Among them,

{\hat{x}}_{1}

{\hat{x}}_{2}

, and

{\hat{x}}_{3}

are the unit direction vectors of the x₁-axis, x₂-axis, and x₃-axis in the induction principal axis coordinate system x₁x₂x₃, respectively.

When the modulation mode of the LiNbO₃ crystals assumed is “the optical axis of the crystal is perpendicular to the direction of light”, after the voltage is applied at both ends of the crystal, it remains uniaxial. The direction of the optical axis of the crystal remains unchanged, but the refractive index inside the crystal changes. The three-dimensional coordinate system of the LiNbO₃ crystal is shown in Figure 3 below.

In Figure 3, O-xyz is the geometric coordinate system of the crystal; the optical axis direction of the crystal coincides with the y-axis direction, and the light direction follows the z-axis direction. When voltage is applied at both ends of the crystal, the inductive spindle coordinate system x₁x₂x₃ of the crystal coincides with X₀Y₀Z₀. The relationship between the sensing principal coordinate system x₁x₂x₃ and the geometric coordinate system xyz can be expressed as follows:

(\begin{matrix} {\hat{x}}_{1} & {\hat{x}}_{2} & {\hat{x}}_{3} \end{matrix}) = (\begin{matrix} \hat{x} & \hat{y} & \hat{z} \end{matrix}) [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}] .

(18)

This paper constructs a three-dimensional coordinate system under two modulation modes for LiNbO₃ crystals, laying a foundation for describing the crystal’s modulation characteristics, which is convenient for analyzing the electro-optical modulation characteristics of LiNbO₃ crystals.

3.2. Birefringence Effect of LiNiO₃ Crystals

Based on Huygens’ principle, light enters an isotropic medium from an isotropic medium, resulting in a birefringence effect. LiNbO₃ crystals are uniaxial crystals in their natural state (when no voltage is added). When the incident light is incident with air to the uniaxial crystal, it will be refracted into two beams of light, one of which is called ordinary light, denoted as O-light, and its refractive index is n_o. The other beam is an anomalous light, called E-light, with a refractive index of n_e. When a voltage is applied to the LiNbO₃ crystal, the crystal transforms into a biaxial crystal. In this case, when the incident light enters the biaxial crystal, two abnormal beams of light are refracted, called fast and slow beams. In this paper, for expression and analysis, these two beams of light are represented by a-light and b-light, respectively. According to the electromagnetic field theory, the phase of light before and after entering the isotropic medium is the same, which can be expressed by the following equation:

{\begin{matrix} n_{i} k_{i} \cdot \hat{x} = n k \cdot \hat{x} \\ n_{i} k_{i} \cdot \hat{y} = n k \cdot \hat{y} \end{matrix},

(19)

where n_i and n are the refractive indices of incident and refracted light, respectively, and k is the vector of refracted light.

In the crystal geometric coordinate system O-xyz, when an incident ray k_i enters the LiNbO₃ crystal from the air, the angle of incidence of the ray is α, and the azimuth angle is ϕ. As shown in Figure 4, the incident ray k_i is refracted on the crystal surface into a-light and b-light, and the wave normal vector of a-light is k_a, which is used to describe the phase propagation direction of light. Similarly, the wave normal vector of b-light is k_b.

Assuming that the incident ray vector k_i is represented by Equation (20) in the three-dimensional coordinate system, the three-dimensional coordinates of the refracted light vector can be obtained by bringing the incident ray vector k into Equation (19), as follows:

k = (\begin{matrix} \hat{x} & \hat{y} & \hat{z} \end{matrix}) [\begin{array}{l} n_{i} \sin α \cos ϕ / n \\ n_{i} \sin α \sin ϕ / n \\ \sqrt{1 - n_{i}^{2} \sin^{2} α / n^{2}} \end{array}],

(20)

where

\hat{x}

\hat{y}

, and

\hat{z}

are the unit direction vectors of the three coordinate axes of the LiNbO₃ crystal in the O-xyz coordinate system, n_i is the refractive index of light in the incident medium, and n is the refractive index of refracted light in the LiNbO₃ crystal.

In order to describe the birefringence effect of LiNbO₃ crystals, the crystal’s refractive index needs to be calculated. According to Maxwell’s equation and the refractive index ellipsoid equation, the Fresnel equation for light in an anisotropic medium can be expressed as follows:

\frac{{k_{x}}^{2}}{\frac{1}{n^{2}} - \frac{1}{n_{1}^{2}}} + \frac{{k_{y}}^{2}}{\frac{1}{n^{2}} - \frac{1}{n_{2}^{2}}} + \frac{{k_{z}}^{2}}{\frac{1}{n^{2}} - \frac{1}{n_{3}^{2}}} = 0,

(21)

where k_x, k_y, and k_z are the components of the refracted light vector in the direction of the refractive principal axis. n is the refractive index of refracted light, and n₁, n₂, and n₃ are the principal refractive index of the LiNbO₃ crystals. Bringing (20) into (21) yields the refractive index n of the ray.

Through the previous explanation of the modulation principle of LiNbO₃ crystals, the refractive index of LiNbO₃ crystals needs to be calculated according to the two modulation methods.

(1): The optical axis of the crystal is parallel to the direction of the light.

When the modulation mode of the crystal is to pass light along the optical axis, the voltage at both ends of the crystal changes from a uniaxial crystal to a biaxial crystal, and the optical axis of the crystal rotates by 45°. By introducing the refractive index Equation (11), the coordinate system transformation Formula (19), and Equation (20), describing the refractive index principal axis of LiNbO₃ after charging into Equation (24), the system of equations regarding the refractive index can be obtained as follows:

A n^{4} - B n^{2} + C = 0 .

(22)

Parameters A, B, and C are expressed as follows:

A = n_{e}^{2},

(23)

\begin{matrix} B = 2 n_{e}^{2} (n_{o}^{2} + n_{E}^{2}) - 2 n_{o} n_{E} n_{i}^{2} \sin^{2} α \cos (2 β) \\ - n_{i}^{2} \sin^{2} α [n_{o}^{2} + n_{E}^{2} - n_{e}^{2}], \end{matrix}

(24)

\begin{matrix} C = {(n_{o}^{2} - n_{E}^{2})}^{2} n_{e}^{2} - 2 n_{o} n_{E} n_{i}^{2} n_{e}^{2} \sin^{2} α \cos (2 β) \\ + n_{i}^{2} \sin^{2} α [(n_{o}^{2} + n_{E}^{2}) n_{e}^{2} - {(n_{o}^{2} - n_{E}^{2})}^{2}] . \end{matrix}

(25)

Solving Equation (22) provides two roots of the refractive index, i.e., the refractive indices of the birefringence a-light and b-light of LiNbO₃ crystals, as follows:

n_{a} = \sqrt{\frac{B + \sqrt{B^{2} - 4 A C}}{2 A}},

(26)

n_{b} = \sqrt{\frac{B - \sqrt{B^{2} - 4 A C}}{2 A}},

(27)

where

n_{E} = \frac{1}{2} n_{o}^{3} γ_{22} \frac{V}{d}

, γ₂₂ is the electro-optical coefficient of the LiNbO₃ crystal.

(2): The optical axis of the crystal is perpendicular to the direction of the light.

When the modulation mode of the crystal is perpendicular to the optical axis, the optical symmetry of LiNbO₃ remains unchanged, and it is still a uniaxial crystal. Equations (15), (18) and (20) are brought into Equation (21) to obtain the refractive indices n_a and n_b of two refracted rays in the LiNbO₃ crystal after an electric field is applied, as follows:

n_{a} = D,

(28)

n_{b} = \sqrt{E^{2} + \frac{[D^{2} - E^{2}] n_{i}^{2} \sin^{2} α \sin^{2} β}{D^{2}}} .

(29)

Parameters D and E are expressed as follows:

D = n_{o} - \frac{1}{2} n_{o}^{3} γ_{13} \frac{V}{d},

(30)

E = n_{e} - \frac{1}{2} n_{e}^{3} γ_{33} \frac{V}{d} .

(31)

By bringing the refractive indices n_a and n_b of the a- and b-light in LiNbO₃ crystals into Equation (20), the refracted light wave vectors k_a and k_b can be obtained, as follows:

k_{a} = (\begin{matrix} \hat{x} & \hat{y} & \hat{z} \end{matrix}) [\begin{array}{l} n_{i} \sin α \cos ϕ / n_{a} \\ n_{i} \sin α \sin ϕ / n_{a} \\ \sqrt{1 - n_{i}^{2} \sin^{2} α / n_{a}^{2}} \end{array}],

(32)

k_{b} = (\begin{matrix} \hat{x} & \hat{y} & \hat{z} \end{matrix}) [\begin{array}{l} n_{i} \sin α \cos ϕ / n_{b} \\ n_{i} \sin α \sin ϕ / n_{b} \\ \sqrt{1 - n_{i}^{2} \sin^{2} α / n_{b}^{2}} \end{array}] .

(33)

Figure 5 depicts a schematic diagram of the propagation of anomalous light within a LiNbO₃ crystal, assuming that the optical paths of the two refracted light wave vectors k_a and k_b in the crystal are La and Lb, since the length of the LiNbO₃ crystal in the clear direction is L. The projection of the optical path La and Lb in the z-axis direction of the crystal is the length of the clear direction L of the crystal; the relationship between the optical path of the refracted light wave vector and the length of the clear direction can be expressed as follows:

L = L a (k_{a} \cdot \hat{z}),

(34)

L = L b (k_{b} \cdot \hat{z}) .

(35)

Assuming that the coordinates of the incident light k_i on the surface of the LiNbO₃ crystal incident point O are (x₀, y₀, z₀), then the coordinates of the exit point of the two refracted light vectors k_a and k_b can be expressed as follows:

(x_{a}, y_{a}, z_{a}) = (x_{o} + L \frac{(k_{a} \cdot \hat{x})}{(k_{a} \cdot \hat{z})}, y_{o} + L \frac{(k_{a} \cdot \hat{y})}{(k_{a} \cdot \hat{z})}, z_{o} + L),

(36)

(x_{b}, y_{b}, z_{b}) = (x_{o} + L \frac{(k_{b} \cdot \hat{x})}{(k_{b} \cdot \hat{z})} y_{o} + L \frac{(k_{b} \cdot \hat{y})}{(k_{b} \cdot \hat{z})}, z_{o} + L) .

(37)

According to the coordinates of the exit points of the two refracted light vectors k_a and k_b, the Euclidean distance between the two exit points can be calculated as follows:

D_{a b} = \sqrt{{(x_{a} - x_{b})}^{2} + {(y_{a} - y_{b})}^{2} + {(z_{a} - z_{b})}^{2}} .

(38)

The optical path length of the two refracted light vectors k_a and k_b within a LiNbO₃ crystal can be expressed as follows:

O L a = \frac{2 π n}{λ} L_{a} = \frac{2 π L a}{λ} n_{a} \frac{1}{(k_{a} \cdot \hat{z})},

(39)

O L b = \frac{2 π n}{λ} L_{b} = \frac{2 π L b}{λ} n_{b} \frac{1}{(k_{b} \cdot \hat{z})} .

(40)

By calculation, the refractive index (n_a, n_b), wave normal vector (k_a, k_b), and optical path (OLa, OLb) of LiNbO₃ crystals are all functions of the ϕ of the angle of incidence α and the azimuth. The phase delay caused by the LiNbO₃ crystals under any incident condition can be expressed as follows:

\begin{array}{l} Γ & = \frac{2 π L}{λ} [n_{a} \frac{1}{(k_{a} \cdot \hat{z})} - n_{b} \frac{1}{(k_{b} \cdot \hat{z})}] \\ = \frac{2 π L}{λ} [n_{a} \frac{1}{\sqrt{1 - \sin^{2} α / n_{a}^{2}}} - n_{b} \frac{1}{\sqrt{1 - \sin^{2} α / n_{b}^{2}}}] . \end{array}

(41)

According to Equation (41), the phase delay caused by the birefringence effect of LiNbO₃ crystals is mainly related to the angle of incidence (angle of incidence α and azimuth angle ϕ), the wavelength λ of the incident light, the length of the direction L, and the refractive index of the refracted light.

4. Analysis of Modulation Characteristics of LiNbO₃ Crystals

This section mainly analyzes the changes in the modulation parameters of LiNbO₃ crystals under the conditions of incidence at any angle of incidence and azimuth, such as refractive index and phase delay, analyzes the modulation law of electro-optical crystals, and explores the modulation performance of these crystals.

4.1. Modulation Propagation Characteristics of LiNbO₃ Crystals

When the electro-optical modulation mode of LiNbO₃ crystals assumed is “the optical axis of the crystal is parallel to the direction of clear light”, the voltage of the LiNbO₃ crystal is applied along the X₀ axis, and the uniaxial crystal changes from a biaxial crystal. The change in the refractive index due to the applied voltage can be expressed as follows:

n_{E} = \frac{1}{2} n_{o}^{3} γ_{22} \frac{V}{d},

(42)

where n_o is the refractive index of ordinary light of LiNbO₃ crystals in the natural state, γ₂₂ is the electro-optical coefficient of the crystals, V is the voltage applied at both ends of the crystals, and d is the length of the voltage applied to the crystals.

When a ray of light is incident perpendicular to a LiNbO₃ crystal, the angle of incidence α and azimuth ϕ of the ray are both equal to 0, at which point the phase delay between the two refracted rays resulting from the birefringence effect can be expressed as follows:

Γ_{α = 0} = \frac{2 π}{λ} n_{o}^{3} γ_{22} \frac{V}{d} L .

(43)

Bringing the condition that the external electric field is 0 into Equations (26) and (27), we can obtain the expression of the refractive index of the LiNbO₃ crystal at any angle of incidence in the uniaxial state, as follows:

n_{A} = n_{o},

(44)

n_{B} = \frac{\sqrt{{n_{e}}^{2} n_{o}^{2} - (n_{o}^{2} - {n_{e}}^{2}) n_{i}^{2} \sin^{2} α}}{n_{e}} .

(45)

As can be seen from the above equation, the refractive index of a-light is constant and does not change with the change of angle of incidence and azimuth. The refractive index of b-light is related to the angle of incidence of the light ray, not the angle of refraction. The phase delay at this point can be expressed as follows:

{Γ |}_{V = 0} = \frac{2 π n_{o} L}{λ} (\frac{n_{o}}{\sqrt{n_{o}^{2} - n_{i}^{2} \sin^{2} α}} - \frac{n_{e}}{\sqrt{{n_{e}}^{2} - n_{i}^{2} \sin^{2} α}}) .

(46)

When the phase delay

Γ_{α = 0} = π

between the two refracted rays, the modulated voltage is called the half-wave voltage of the LiNbO₃ crystal, and it can be expressed as follows:

V_{π} = \frac{λ}{2 n_{o}^{3} γ_{22}} \frac{d}{L} .

(47)

λ is the wavelength of the incident light, and the half-wave voltage modulated by the crystal is mainly related to the five parameters in Equation (47). Among them, no and γ₂₂ are the performance parameters determined by the crystal’s internal structure, d and L are the parameters that describe the crystal size, and the larger the crystal aperture (d), the higher the modulation half-wave voltage. According to the basic parameters of LiNbO₃ crystals provided in Table 2, the half-wave voltage V_π = 2011.35V is calculated.

When the electro-optical modulation mode of LiNbO₃ crystals assumed is “the optical axis of the crystal is perpendicular to the direction of clear light”, the voltage of the LiNbO₃ crystal is applied along the Z₀ axis, and the LiNbO₃ crystal is still uniaxial after the voltage is applied. The change in the refractive index due to the applied voltage can be expressed as follows:

\begin{array}{l} n_{E 1} = \frac{1}{2} n_{o}^{3} γ_{13} \frac{V}{d}, \\ n_{E 2} = \frac{1}{2} {n_{e}}^{3} γ_{33} \frac{V}{d} . \end{array}

(48)

Among them, n_o and n_e are the refractive indices of the ordinary light and abnormal light of LiNbO₃ crystals in the natural state, γ₁₃ and γ₃₃ are the electro-optical coefficients of the crystals, V is the voltage applied at both ends of the crystals, and d is the length of the voltage direction applied to the crystals.

Based on the amount of refractive index change caused by the voltage applied to the LiNbO₃ crystal, the phase delay between the refracted rays when the light rays are incident perpendicular to the LiNbO₃ crystal can be calculated as follows:

Γ_{α = 0} = \frac{2 π}{λ} (n_{o} - n_{e}) L + \frac{π L}{λ} \frac{V}{d} (n_{e}^{3} γ_{33} - n_{o}^{3} γ_{13}) .

(49)

Compared with the modulation of the crystal optical axis parallel to the clear light direction, the phase delay here is not only caused by the electro-optical modulation but also by the delay term introduced by the crystal’s natural birefringence.

When the phase delay

Γ_{α = 0} = π

between the two refracted rays, the modulated voltage is called the half-wave voltage of the LiNbO₃ crystal, and when the optical axis of the crystal is perpendicular to the direction of the light, the half-wave voltage can be expressed as follows:

V_{π} = λ d / [L (n_{e}^{3} γ_{33} - n_{o}^{3} γ_{13})] .

(50)

According to the basic parameters of the LiNbO₃ crystal provided in Table 2, the half-wave voltage V_π = 1429.66 V is calculated.

4.2. LiNbO₃ Crystal Modulated Light Intensity and Field of View Analysis

When the angle of incidence of the ray is α, and the azimuth angle is ϕ, the phase difference

θ (α, β)

of the birefringent rays k_a and k_b can be calculated according to Equation (41), as follows:

\begin{array}{l} θ (α, β) & = \frac{2 π L}{λ} [n_{a} \frac{1}{(k_{a} \cdot \hat{z})} - n_{b} \frac{1}{(k_{b} \cdot \hat{z})}] \\ = \frac{2 π L}{λ} [n_{a} \frac{1}{\sqrt{1 - \sin^{2} α / n_{a}^{2}}} - n_{b} \frac{1}{\sqrt{1 - \sin^{2} α / n_{b}^{2}}}] . \end{array}

(51)

After obtaining the phase difference of the refracted ray, in order to calculate the interference fringes, it is necessary to obtain the polarization direction

ϕ (α, β)

of the refracted ray. The direction of polarization of the refracted rays is determined with the help of a refractive index ellipsoid, shown in Figure 6. An ellipsoid has two circular sections C₁ and C₂ passing through the center of the sphere, the normal ranges of these two sections N₁ and N₂, called the optical axis of the crystal, are in the X’OZ’ plane, and in the O-XYZ coordinate system, the unit normal is

ρ = {(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 0)}^{T}

of the optical axis surface. For ray k propagating in a non-optical axis direction, the center of the refractive index ellipsoid is a plane Q perpendicular to k, and its cross-section with the ellipsoid is an ellipse. The plane Q intersects the circles C₁ and C₂ at r₁ and r₂, and the length of the two is equal, so the principal axis direction of the ellipse Q is the direction of the angular bisector of r₁ and r₂, which is also the direction of polarization E of the ray k.

The 3D coordinates of the unit direction vectors of the optic axes N₁ and N₂ are denoted as

N_{1} = (\cos η, 0, \sin η)

and

N_{2} = (\cos η, 0, - \sin η)

, respectively.

η

is the angle between the optic axis and the Z-axis, which can be expressed as follows:

η = \arctan (\frac{n_{1}}{n_{3}} \sqrt{\frac{{n_{2}}^{2} - {n_{3}}^{2}}{{n_{1}}^{2} - {n_{2}}^{2}}}) .

(52)

Since r₁ is perpendicular to k and N₁, r₁ can be calculated, and in a similar way, r₂ can also be calculated; the formula can be expressed as follows:

{\begin{matrix} r_{1} = \frac{k \times N_{1}}{| k \times N_{1} |} \\ r_{2} = \frac{k \times N_{2}}{| k \times N_{2} |} \end{matrix} .

(53)

The polarization direction of the light rays is

E = \frac{1}{2} (r_{1} + r_{2})

. The angle between the polarization direction and the optical axis surface can be expressed as follows:

ϕ (α, β) = \arcsin (\frac{E \cdot ρ}{| E | | ρ |}) .

(54)

Knowing the phase difference

θ (α, β)

and the direction of polarization

ϕ (α, β)

, the light intensity formula can be expressed as follows:

I (α, β) = \sin^{2} [2 ϕ (α, β)] \sin^{2} [\frac{θ (α, β)}{2}] .

(55)

According to Equation (55), the interference pattern of the crystal calculates the time, and the simulation results are shown in Figure 7 below.

Figure 7a shows the distribution of light intensity after light passes through the crystal when the voltage V = 0 at both ends of the crystal, and the field angle of view is 4°. Figure 7b,c shows the influence of polarization direction on light intensity

\sin^{2} [2 ϕ (α, β)]

and the effect of phase difference on light intensity

\sin^{2} [\frac{θ (α, β)}{2}]

, respectively. The field of view in (a), (b) and (c) was enlarged 1° to obtain Figure 7d–f. As can be seen from Figure 7d, the uniformity of the image center is good, and the image uniformity is poor at the edge of the field of view in the azimuth angles 45°, 135°, 225°, and 315°. This is due to the fact that the interference light intensity is composed of two parts: the homochromic line (shown in Figure 7b) and the isochromatic line (shown in Figure 7c). The homochromic line displays bright areas in the four directions of azimuth angles 45°, 135°, 225°, and 315°. Through the analysis of the simulation data, the light intensity in the 1° field of view in Figure 7d is less than 0.07, and it is considered that the central field of view is in a dark state at this time.

The voltage at both ends of the crystal is set to

V = V_{π}

during the simulation in Figure 7, and the simulation results are shown in Figure 8.

Figure 8d shows that the light intensity within 1° of the field of view is greater than 0.87, and the electro-optical modulation is considered to be bright at this time. However, the aberration at the edge of the field of view is large, and the source of the aberration is analyzed according to the interference intensity formula. In Figure 8e, the range of light intensity is 0.95–1; Figure 8f shows that the range of light intensity is 0.86–1. Therefore, the polarization direction

ϕ (α, β)

has little effect on the light intensity within the 1° field of view, and the non-uniformity of the light intensity is mainly related to the phase difference

θ (α, β)

Through the analysis of the propagation characteristics of LiNbO₃ crystals, the effective field of view of the LiNbO₃ crystals selected in this paper is about 1° for electro-optical modulation. At a field of view of 1°, the uniformity of the crystal is better. It should be noted that the field of view of a crystal is related to its size, so in practical applications, the appropriate crystal should be selected for electro-optical modulation according to different application scenarios.

The effect of temperature on light intensity is more significant when it comes to optical materials such as LiNbO₃ crystals. On the one hand, the thermo-optical effect of LiNbO₃ crystals will change the refractive index along with temperature change, affecting the propagation path and intensity of light in the crystal. Thermal expansion can also change the crystal size due to temperature changes, affecting optical parameters and light intensity. At the same time, temperature also has an effect on the output light intensity and spectral characteristics of emitting light sources such as semiconductor lasers. To improve the results, temperature control can be employed, such as the use of an incubator or thermoelectric cooler. Material selection and optimization, such as the selection of thermally stable materials or the optimization of the LiNbO₃ crystal growth process, can also be employed. Optical design optimization can also be carried out, compensation or adaptive design methods can be adopted, and the structure and parameters of the optical system can be optimized to reduce the influence of temperature on light intensity and improve the performance and stability of the optical system.

5. Conclusions

Based on the refractive index ellipsoid theory, the propagation characteristics of polarized light in LiNbO₃ crystals are studied using the three-dimensional ray tracing method, and the optical properties of LiNbO₃ crystals under different conditions are analyzed by establishing a refractive index ellipsoid analysis model. The model accurately describes the refractive index changes of crystals, providing a powerful tool for studying the birefringence and Pockels effect of crystals. The Pockels effect on the model LiNbO₃ crystals was studied in detail, revealing the variation of the optical properties of the crystal under the action of an external electric field. Secondly, this paper clarifies the difference in the optical characteristics of the crystal when the optical axis is parallel to the clear direction and perpendicular to the clear direction, which provides an important basis for the design and optimization of electro-optical modulation. Finally, we propose a three-dimensional ray tracing method for electro-optical crystals and construct a three-dimensional coordinate system. This method can accurately simulate the propagation path of light in the crystal, which provides an effective means to study the modulation propagation characteristics of the crystal. The modulated light intensity and field of view of the electro-optical crystal were analyzed. The relationship between the light intensity and field angle and the crystal parameters, electric field strength, and other factors was clarified. The effective field of view of the LiNbO₃ crystals selected in this paper was about 1°. This provides a basis for the design of electro-optical modulation devices with appropriate light intensities and fields of view. This series of research results not only enriches the understanding of the electro-optical characteristics of LiNbO₃ crystals but also provides an important theoretical basis and practical guidance for designing and optimizing electro-optical modulation devices based on LiNbO₃ crystals. Electro-optical crystal modulation has great application prospects in the fields of high-speed signal modulation, optical switching and routing, Q-switched switching, and microwave signal generation and processing. Future research can be extended to different types of electro-optical crystals and more complex optical systems to promote the continuous development of optoelectronic technology.

6. Patents

The authors have applied for national invention patents for the relevant content of the article under the following patent name: a crystal electro-optic modulation method based on three-dimensional ray tracing, patent number: ZL 2023 1 0450190.6.

Author Contributions

All authors carried out and analyzed all experiments. H.S. wrote the manuscript, which all authors discussed. Conceptualization, H.S. and Y.L.; methodology, H.S.; software, H.S.; validation, L.Z., H.G. and H.Z.; formal analysis, R.L.; investigation, Y.F.; resources, H.S.; data curation, H.S.; writing—original draft preparation, H.S.; writing—review and editing, C.L.; visualization, H.S.; supervision, H.S.; project administration, H.S.; funding acquisition, H.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data that support this study are proprietary and may only be provided with restrictions.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The refractive index ellipsoid of a crystal.

Figure 2. Cartesian coordinate system of LiNbO₃ crystals when the optical axis of the crystal is parallel to the direction of clear light.

Figure 3. Cartesian coordinate system of LiNbO₃ crystals when the optical axis of the crystal is parallel to the direction of clear light.

Figure 4. Birefringence at the interface between an isotropic medium and a LiNbO₃ crystal.

Figure 5. Propagation of abnormal light in LiNiO₃ crystals.

Figure 6. Refractive index ellipsoid.

Figure 7. Crystal conoscopic interferogram when V = 0: (a) interference light intensity distribution; (b) influence of polarization direction on light intensity

\sin^{2} [2 ϕ (α, β)]

; (c) influence of phase difference on light intensity

\sin^{2} [\frac{θ (α, β)}{2}]

; (d–f) is a magnification of the field of view at the center of (a–c) of the figure.

Figure 7. Crystal conoscopic interferogram when V = 0: (a) interference light intensity distribution; (b) influence of polarization direction on light intensity

\sin^{2} [2 ϕ (α, β)]

; (c) influence of phase difference on light intensity

\sin^{2} [\frac{θ (α, β)}{2}]

; (d–f) is a magnification of the field of view at the center of (a–c) of the figure.

Figure 8. Crystal conoscopic interferogram when V = V_π: (a) interference light intensity distribution; (b) influence of polarization direction on light intensity

\sin^{2} [2 ϕ (α, β)]

; (c) influence of phase difference on light intensity

\sin^{2} [\frac{θ (α, β)}{2}]

; (d–f) is a magnification of the field of view at the center of (a–c) of the figure.

Figure 8. Crystal conoscopic interferogram when V = V_π: (a) interference light intensity distribution; (b) influence of polarization direction on light intensity

\sin^{2} [2 ϕ (α, β)]

; (c) influence of phase difference on light intensity

\sin^{2} [\frac{θ (α, β)}{2}]

; (d–f) is a magnification of the field of view at the center of (a–c) of the figure.

Table 1. Table of refractive index of various crystal group spindles.

Crystal Family	Principal Refractive Index Relationship	Included Crystal Systems	Crystal Name
Advanced crystal family	$n_{X 0} = n_{Y 0} = n_{Z 0}$	Cubic System	Isotropic crystals
Intermediate crystal family	$n_{X 0} = n_{Y 0} \neq n_{Z 0}$ or $n_{X 0} \neq n_{Y 0} = n_{Z 0}$ or $n_{X 0} = n_{Z 0} \neq n_{Y 0}$	Trigonal System; Tetragonal System; Hexagonal System	Uniaxial crystal
Lower crystal families	$n_{X 0} \neq n_{Y 0} \neq n_{Z 0}$	Triclinic System; Monoclinic System; Orthorhombic System	Biaxial crystals

Table 2. Performance parameters of LiNbO₃ crystals.

Parameter	Numeric Value
The refractive index of ordinary O-light in its natural state is n_o.	2.2797
The refractive index of ordinary E-light in its natural state is n_e.	2.1969
The length L in the direction of the light	18.8 mm
Crystal thickness d in the direction of applied voltage	9 mm
Electro-optic coefficient $γ_{22}$	6.74 × 10⁻¹² m/V
Electro-optic coefficient $γ_{13}$	8.6 × 10⁻¹² m/V
Electro-optic coefficient $γ_{33}$	30.8 × 10⁻¹² m/V

Table 3. Optical properties of LiNbO₃ crystals under the Pockels effect.

Power-Up Direction	Direction of Light Transmission	Optical Symmetry	Spindle Rotation Characteristics	Principal Refractive Index	The Amount of Change in the Refractive Index
X₀ axis	Z₀ axis	Biaxial crystals	θ = 45° (Z₀ axis)	$\begin{array}{l} n_{1} = \frac{1}{n_{o}^{2}} - γ_{22} E_{X 0} \\ n_{2} = \frac{1}{n_{o}^{2}} + γ_{22} E_{X 0} \\ n_{3} = n_{e} \end{array}$	$n_{E} = \frac{1}{2} n_{o}^{3} γ_{22} E_{X 0}$
Y₀ axis	Z₀ axis	Biaxial crystals	Does not rotate	$\begin{array}{l} n_{1} = \frac{1}{n_{o}^{2}} + γ_{22} E_{X 0} \\ n_{2} = \frac{1}{n_{o}^{2}} - γ_{22} E_{X 0} \\ n_{3} = n_{e} \end{array}$	$n_{E} = \frac{1}{2} n_{o}^{3} γ_{22} E_{Y 0}$
Z₀ axis	X₀ axis or Y₀ axis	Uniaxial crystal	Does not rotate	$\begin{array}{l} n_{1} = n_{o} - \frac{1}{2} n_{o}^{3} γ_{13} E_{Z 0} \\ n_{2} = n_{o} - \frac{1}{2} n_{o}^{3} γ_{13} E_{Z 0} \\ n_{3} = n_{e} - \frac{1}{2} n_{e}^{3} γ_{33} E_{Z 0} \end{array}$	$\begin{array}{l} n_{E 1} = \frac{1}{2} n_{o}^{3} γ_{13} E_{Z 0} \\ n_{E 2} = \frac{1}{2} {(n_{o} - n_{Δ})}^{3} γ_{33} E_{Z 0} \end{array}$

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Sun, H.; Li, Y.; Zhang, L.; Guo, H.; Luan, C.; Zheng, H.; Li, R.; Fan, Y. Research on the Modulation Characteristics of LiNbO₃ Crystals Based on the Three-Dimensional Ray Tracing Method. Crystals 2024, 14, 1101. https://doi.org/10.3390/cryst14121101

AMA Style

Sun H, Li Y, Zhang L, Guo H, Luan C, Zheng H, Li R, Fan Y. Research on the Modulation Characteristics of LiNbO₃ Crystals Based on the Three-Dimensional Ray Tracing Method. Crystals. 2024; 14(12):1101. https://doi.org/10.3390/cryst14121101

Chicago/Turabian Style

Sun, Houpeng, Yingchun Li, Laixian Zhang, Huichao Guo, Chenglong Luan, Haijing Zheng, Rong Li, and Youchen Fan. 2024. "Research on the Modulation Characteristics of LiNbO₃ Crystals Based on the Three-Dimensional Ray Tracing Method" Crystals 14, no. 12: 1101. https://doi.org/10.3390/cryst14121101

APA Style

Sun, H., Li, Y., Zhang, L., Guo, H., Luan, C., Zheng, H., Li, R., & Fan, Y. (2024). Research on the Modulation Characteristics of LiNbO₃ Crystals Based on the Three-Dimensional Ray Tracing Method. Crystals, 14(12), 1101. https://doi.org/10.3390/cryst14121101

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Research on the Modulation Characteristics of LiNbO₃ Crystals Based on the Three-Dimensional Ray Tracing Method

Abstract

1. Introduction

2. Electro-Optical Effect of LiNbO₃ Crystals

2.1. Refractive Index Ellipsoid Analytical Model

2.2. The Pockels Effect of LiNbO₃ Crystals

3. Three-Dimensional Ray Tracing Method for LiNbO₃ Crystal Modulation

3.1. The 3D Coordinate System Construction

3.2. Birefringence Effect of LiNiO₃ Crystals

4. Analysis of Modulation Characteristics of LiNbO₃ Crystals

4.1. Modulation Propagation Characteristics of LiNbO₃ Crystals

4.2. LiNbO₃ Crystal Modulated Light Intensity and Field of View Analysis

5. Conclusions

6. Patents

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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Research on the Modulation Characteristics of LiNbO3 Crystals Based on the Three-Dimensional Ray Tracing Method

Abstract

1. Introduction

2. Electro-Optical Effect of LiNbO3 Crystals

2.1. Refractive Index Ellipsoid Analytical Model

2.2. The Pockels Effect of LiNbO3 Crystals

3. Three-Dimensional Ray Tracing Method for LiNbO3 Crystal Modulation

3.1. The 3D Coordinate System Construction

3.2. Birefringence Effect of LiNiO3 Crystals

4. Analysis of Modulation Characteristics of LiNbO3 Crystals

4.1. Modulation Propagation Characteristics of LiNbO3 Crystals

4.2. LiNbO3 Crystal Modulated Light Intensity and Field of View Analysis

5. Conclusions

6. Patents

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

Share and Cite

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Further Information

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Research on the Modulation Characteristics of LiNbO₃ Crystals Based on the Three-Dimensional Ray Tracing Method

2. Electro-Optical Effect of LiNbO₃ Crystals

2.2. The Pockels Effect of LiNbO₃ Crystals

3. Three-Dimensional Ray Tracing Method for LiNbO₃ Crystal Modulation

3.2. Birefringence Effect of LiNiO₃ Crystals

4. Analysis of Modulation Characteristics of LiNbO₃ Crystals

4.1. Modulation Propagation Characteristics of LiNbO₃ Crystals

4.2. LiNbO₃ Crystal Modulated Light Intensity and Field of View Analysis