An Introduction to Optical Crystal Structures

An Introduction to Optical Crystal Structures

CASTECH is a leading global supplier of crystal components for the laser industry. With over 30 years of dedicated development, we have established a comprehensive product line of optical crystals, including nonlinear optical, laser, acousto-optic, electro-optic, magneto-optic, and birefringent crystals, all supported by our expert technical services. In particular, our market share for crystals such as LBO, BBO, and Nd:YVO₄ has consistently led the world.

Through long-term collaboration and mutual growth with our customers, we understand that everything—from optimizing crystal growth on the supply side to designing and implementing them in application systems on the user end—depends on a fundamental grasp of the crystal's underlying microscopic structure and its manifest physical and chemical properties.

So, what are the structural characteristics of these optical crystals? And how can we leverage these characteristics to use them more effectively? In today's Cas’Tech Class, we will explore these questions together.

1. Fundamental Concepts

The most fundamental structural characteristic of a crystal is the three-dimensional, periodic, and orderly arrangement of its internal atoms, ions, or molecules. Figure 1 shows the unit cell structure of an LBO crystal.

The smallest structural unit of a crystal is the unit cell. The directions of the three edges of the unit cell are called the crystallographic axis, typically denoted as a (X), b (Y), and c (Z). For trigonal and hexagonal crystal systems, an additional horizontal axis is introduced, resulting in three horizontal axis oriented at 120° to each other.

Figure 1. The unit cell structure of an LBO crystal

According to their optical properties, crystals can be divided into two broad categories:

• Optically homogeneous (isotropic) materials

• Optically heterogeneous (anisotropic) materials

Isotropic crystals always have a cubic unit-cell structure.

Anisotropic crystals, on the other hand, can belong to any of six unit-cell systems: trigonal, tetragonal, hexagonal, orthorhombic, monoclinic, or triclinic.

• The trigonal, tetragonal, and hexagonal systems are uniaxial crystals, characterized by dielectric constants εx = εy ≠ εz; thus the a and b axis share identical physical properties.

• The orthorhombic, monoclinic, and triclinic systems are biaxial crystals, with εx ≠ εy ≠ εz.

Figure 2. The basic classification framework of crystal structures.

Laser crystals such as Yb:CaF2 and Er:Cr:YSGG, as well as magneto-optical crystals like TGG, belong to the cubic crystal system; nonlinear crystals β-BBO, electro-optic crystals LiNbO3, and laser crystals Cr:LiSAF belong to the trigonal system; nonlinear crystal CLBO, laser crystal Yb:CALGO, and birefringent crystal YVO4 belong to the tetragonal system; nonlinear crystals LBO and KTP belong to the orthorhombic system; and nonlinear crystal BIBO belongs to the monoclinic system, etc.

Why do we perform such a detailed classification of optical crystals? Because the direction of light propagation and polarization within different crystal structures are critical to their applications and performance.

2. The Optical Indicatrix

The propagation of light in isotropic crystals is relatively simple; the speed of propagation does not change with the direction of the light wave's propagation or vibration. In this article, we focus on the anisotropic case and introduce the concept of the optical indicatrix (also called the index ellipsoid) to visually represent the refractive index for light vibrating in a specific direction.

As shown in Figure 3, anisotropic crystals can be divided into uniaxial and biaxial crystals. The optical indicatrix for a uniaxial crystal is an ellipsoid of revolution (spheroid). It is formed with the Ne axis (parallel to the c-axis, with a magnitude equal to the extraordinary refractive index) as the axis of rotation and the No axis (perpendicular to the c-axis, with a magnitude equal to the ordinary refractive index) as the radius. The crystal is classified as having a positive or negative optical sign based on the relative magnitudes of Ne and No, as shown in Figure 4(a) and (b). The optical indicatrix for a biaxial crystal is a triaxial ellipsoid, where Ng, Nm, and Np represent the maximum, intermediate, and minimum principal refractive indices, respectively. The sections of the indicatrix perpendicular to the optic axis are circles with a radius of Nm, as shown in Figure 4(c) and (d).

Figure 3. The classification of anisotropic crystals

1. Positive uniaxial crystal                  (b)Negative uniaxial crystal

1. Positive biaxial crystal                  (d)Negative biaxial crystal

Figure 4. The optical indicatrix for anisotropic crystals

The optical indicatrix is essential in the design and use of optical crystals, such as in the calculation of phase matching angles and nonlinear coefficients for nonlinear crystals. Below, we will use the applications of the β-BBO crystal in nonlinear optical frequency conversion and electro-optic Q-switching as examples to discuss the specific role of the optical crystal structure, particularly the optical indicatrix.

3. Applications of the Optical Indicatrix

β-BBO is a negative uniaxial crystal belonging to the trigonal system. The boule orientation and the refractive index ellipsoid (optical indicatrix) are shown in Figure 5.

Figure 5. The The boule orientation and the refractive index ellipsoid of BBO crystal

Figure 6 illustrates the relationship between the refractive index surfaces for the two different polarization states of the 1064 nm fundamental wave and the 532 nm second harmonic wave in the BBO crystal. The spherical surface represents the refractive index surface for the o-ray (ordinary light), the ellipsoidal surface represents the refractive index surface for the e-ray (extraordinary light), and z is the optic axis.

Figure 6. The relationship between the refractive index surfaces for the two different polarization states

Due to anisotropy in nonlinear crystals, the ordinary light (o-wave) and extraordinary light (e-wave) exhibit different refractive indices, resulting in the phenomenon of birefringence. The refractive index of the e-wave changes with temperature at a different rate than that of the o-wave, creating the possibility that the participating light waves can propagate through the medium at the same speed, thereby enabling effective frequency conversion.

For uniaxial crystals in the frequency doubling process, one method to achieve phase matching is to find the intersection point between the fundamental light and the frequency-doubled light (see Table 1). From Figure 6, we can observe the intersection point between the fundamental light at n_o (1064 nm) and the frequency-doubled light at n_e (532 nm). Phase matching can be achieved when light propagates along a direction with an angle θ_m relative to the optical axis z. According to calculations, the phase matching angle for β-BBO here is θ = 22.8° and φ = 0° [1]. In the crystal processing step, directional procedures can utilize X-ray orientation devices to determine the phase matching angle surface for the 1064 nm frequency doubling of β-BBO (θ = 22.8°, φ = 0°), and subsequently proceed with cutting, grinding, polishing, and other relevant processes.

Table 1. Phase matching condition for uniaxial crystals

Another important application of β-BBO crystals is the use of the electro-optic effect along the Z-cut (θ = 0°) direction for electro-optic Q-switching. As shown in Figure 7,  the incident light is polarized linearly along the Y-axis. When a voltage is applied to the β-BBO crystal along the Y-axis, the refractive index ellipsoid changes, effectively transforming it from uniaxial to biaxial. As a result, the X and Y axes experience different angular deflections depending on the applied voltage. The changes in the refractive indices of the X' and Y' axes upon applying pressure are as follows:

Figure 7. Principle of β-BBO electro-optic Q-switching

When a λ/4 voltage is applied along the Y-axis, a linearly polarized light beam incident along the Y-axis decomposes into two polarization components aligned with the X′-axis and Y′-axis within the crystal. This results in birefringence, where the ordinary ray (o-ray) and extraordinary ray (e-ray) acquire a π/2 phase difference. Consequently, the emergent light becomes circularly polarized. When a λ/2 voltage is applied along the Y-axis, the o-ray and e-ray accumulate a π phase difference, causing the emergent light to be linearly polarized with its polarization plane rotated by 90° relative to the incident light. Thus, the electro-optic crystal functions as an optical waveplate with an adjustable phase retardation.

References‌

‌[1]‌ Yao Jianquan. Nonlinear Optical Frequency Conversion and Laser Tuning Technology [M]. Beijing: Science Press, 1995.

‌[2]‌ Xu Ziyi. Research on High-Repetition-Rate Electro-Optic Q-Switched Lasers [D]. Huazhong University of Science and Technology, 2011.T