The variable-divergence Galilean optical system is an afocal configuration capable of continuously adjusting the divergence angle of a laser beam through axial translation of its internal lens groups [1][2][14]. The classical Galilean architecture typically consists of three optical groups arranged in either a negative-negative-positive or positive-negative-positive configuration [1][2][15]. In this study, the latter configuration is adopted, consisting of three lens groups defined as follows: L 1 —the front fixed positive group, L 2 —the movable negative zoom group, and L 3 —the positive compensating group. The effective focal lengths of the three groups are denoted as f ′ 1 , f ′ 2 , and f ′ 1 , respectively, while the initial axial separations between them are represented by d 12 and d 23 .

During operation, the system continuously maintains the afocal condition, meaning that both the incident and emergent beams remain collimated, while the angular magnification varies with the axial displacement of the internal lens groups. When the zoom group L 2 is translated by an axial distance Δ d 2 , the location of its image plane shifts accordingly, which may cause the system to deviate from its afocal state [1][2]. To restore this condition, the compensating group L 3 must be translated by a corresponding amount Δ d 3 (as illustrated in Figure 1(b)), determined by the lateral magnification m 2 ( Δ d 2 ) of the zoom group [1][2]:

The beam divergence ratio M and the corresponding relationship of the divergence angles are defined as follows [1][2]:

where h 1 and h 3 represent the beam heights at the entrance and exit, respectively.

The set of Equations (1)-(4) describes the axial displacement laws of Δ d 2 and Δ d 3 , which ensure that the system remains in the afocal state throughout all zoom configurations.

Figure 1. Principle layout of the afocal zoom beam expander. (a) System with fixed focus; (b) System with variable focus [1][2].

This analytical model enables rapid determination of the axial positions of the lens groups for any desired magnification and serves as the theoretical foundation for optical optimization in Zemax software. To further enhance the optical performance and manufacturability of the system, the first lens group L 1 is replaced by a NAL.

We consider a thin Kerr medium layer of thickness d , characterized by a linear refractive index n 0 and a nonlinear refractive index coefficient n 2 . When a Gaussian laser beam (TEM₀₀) with peak intensity I 0 and waist radius W 0 propagates through this medium, the refractive index distribution across the transverse (radial) coordinate can be expressed as follows [11]-[13][16]-[20]:

As is well known, the radial refractive index distribution within the Kerr medium induces an optical path difference (OPD) that is equivalent to that produced by an aspherical surface. By expanding the OPD expression and comparing it with the sag function of a rotationally symmetric aspherical surface, the equivalent radius of curvature R at the vertex and the conic coefficient k of the NAL can be derived [11][12].

where, λ denotes the operating wavelength. Accordingly, the NAL can be modeled in Zemax software as an aspherical surface, enabling accurate simulation of light propagation through the nonlinear medium. When the NAL replaces the first group L 1 , the system preserves the classical afocal operating mechanism described by Equations (1)-(4). It should be noted that the focal length f ′ of a conventional lens depends on the surface curvatures R 1 , R 2 and the refractive index n , and is determined by the following relationship [21]-[25]:

Therefore, for the NAL, which is equivalent to an aspherical lens consisting of one planar surface and one aspherical surface, the focal length of the NAL corresponding to the effective focal length f ′ 1 of the first group L 1 is determined as follows:

It is evident that when the laser intensity I 0 varies, the effective focal length f ′ 1 ( I 0 ) of the NAL changes accordingly, together with the radius of curvature R and the conic coefficient k . This nonlinear response modifies the equivalent aspherical profile of the NAL and enables compensation of spherical aberration.

As a result, the incorporation of the NAL improves wavefront quality and helps maintain stable optical performance over the entire zoom range. The combination of axial mechanical translation and nonlinear aberration compensation leads to a high-performance optical system suitable for practical implementation. This configuration demonstrates the feasibility of integrating nonlinear materials into the design of variable divergence laser optical systems, paving the way for the development of adaptive and industrially manufacturable optical architectures.

Based on the theoretical framework discussed above, a variable-divergence Galilean optical system integrating a NAL is proposed and designed according to the optical layout illustrated in Figure 2. Using Zemax software, the system parameters and output beam quality were optimized. The system adheres to the conventional axial-translation principle of the Galilean afocal configuration, as employed in previous designs, but replaces the first lens with a nonlinear element to enhance wavefront quality and reduce manufacturing complexity and cost.

Figure 2.The sketch of afocal zoom beam expander with NAL.

The optical configuration consists of three lens groups: the NAL; L 2 , a movable negative group used to control the beam divergence; and L 3 , a positive compensating group that maintains the afocal condition across the entire zoom range. The designed optical system is required to satisfy the following design specifications [1][2]:

Input beam: Gaussian beam, 1 mm diameter, wavelength 1064 nm;Output beam: continuously variable diameter from 4 mm to 20 mm;Optical quality: wavefront aberration RMS < λ / 14 , free from vignetting;Mechanical structure: total system length approximately 150 mm.

The initial system configuration was constructed based on a conventional variable Galilean design employing spherical lenses. The initial spacings d 12 and d 23 , as well as the displacement values Δ d 2 and Δ d 3 , were determined using Equations (1)-(4). The system was modeled in multi-configuration mode in Zemax software, where d 12 and d 23 were varied to control the relative positions of the lens groups during zooming ( Δ d 2 , Δ d 3 ) . The “Afocal Image Space” mode was activated to ensure that the output beam remained collimated along the optical axis.

The optical design was performed using sequential ray tracing in Zemax OpticStudio. The first lens L 1 was replaced with a NAL, modeled as an equivalent even aspherical surface with curvature radius R and conic coefficient k calculated from Equation (6). The nonlinear medium selected was a thin film of Oil Red O (ORO) [20][26], characterized by a nonlinear refractive index coefficient n₂ = 10⁻⁶ cm²/W, a linear refractive index n = 1.50, a linear absorption coeficient β ≈ 10⁻⁴ W/cm and a film thickness d = 1 mm. The laser beam waist W₀ = 0.5 mm and peak intensity I₀ = 10⁵ W/cm² were also defined for the simulation. All initial design parameters are summarized in Table 1.

Table 1.Design parameters of initial proposal optical system.

Table 2.Laser beam opening angle constraints.

The optimization process was carried out in the multi-configuration mode of Zemax software, where different zoom states were simulated by varying the axial separations between lens groups. The main variables optimized include radii of curvature, conic coefficients, and inter-element spacings.

Four representative configurations were defined to cover the entire zoom range of the system, corresponding to output beam diameters of 2 mm, 5 mm, 8 mm, and 10 mm. The merit function was constructed using REAY operands to control the output beam size, together with constraints on wavefront aberration and afocality. The Physical Optics Propagation (POP) module was subsequently used to verify the wavefront quality. Each REAY term defines the vertical coordinate of the chief ray (Py = 1.0) for a given configuration, corresponding to the required output beam radius. Target values of 2 mm, 5 mm, 8 mm, and 10 mm were assigned respectively to configurations 1-4, ensuring that the system achieves the precise beam divergence across the full zoom range (see Table 2). In addition to the beam-divergence constraints, the merit function also included conditions for controlling the wavefront aberration (RMS < λ/14) and maintaining the afocal condition by enforcing the chief rays to exit the system parallel to the optical axis [9][10][23]. Further constraints were imposed on lens thicknesses and inter-lens spacings to ensure compliance with fabrication tolerances and practical requirements.

The optimization process was iterative and progressively convergent, performed through multiple refinement cycles to ensure that all system parameters satisfied the design requirements for beam divergence, wavefront aberration, and overall optical performance. The resulting variable divergence Galilean optical system integrating the NAL was successfully optimized. The final optical configuration after optimization is illustrated in Figure 3, and the principal optical parameters of the system including lens curvatures, thicknesses, and group separations are summarized in Table 3.

Table 3.Parameters of optical system after optimization.

Figure 3.Optical system configuration after optimization.

From Table 3, it can be seen that the total optical length of the system is less than 150 mm, ensuring that the beam divergence varies with the relative axial displacement between L 2 and L 3 . The NAL functions as an aberration correcting element, effectively reducing the overall optical aberrations of the system. The optimization results indicate that the optical system consists of two aspherical lenses: the NAL and the movable negative lens L 2 . The parameters of both lenses are listed in Table 4. The axial separation d 12 between the NAL and L 2 , and d 23 between L 2 and L 3 , for four different configurations are summarized in Table 5.

The optimized system can vary the diameter of the output laser beam from 4 mm, 10 mm, 16 mm to 20 mm, corresponding to expansion ratios of 4^×, 10^×, 16^×, and 20^×, as illustrated in the footprint diagram (Figure 4). The RMS wavefront aberration for the four configurations is below λ/14, with values of 0.060λ, 0.070λ, 0.036λ, and 0.033λ, respectively. The peak-to-valley (PV) wavefront aberration is also below λ/4, with corresponding values of 0.245λ, 0.211λ, 0.126λ, and 0.242λ. These results demonstrate that the optical performance of the optimized system approaches the diffraction limit, confirming that the design meets the desired optical quality.

Table 4.Parameters of aspherical lenses.

Table 5. Thickness between neighbor moving components.

The obtained results are in good agreement with those of previous systems [1][2][7][15], which employed three aspherical lenses, and even show slightly better performance compared with systems that use two aspherical lenses combined with additional spherical surfaces [1][2][15].

Accordingly, the optimized optical system satisfies the design requirements in terms of both beam divergence and optical quality. This confirms that the incorporation of a nonlinear aspherical lens (NAL) into the variable-divergence Galilean system fully meets the criteria of a high-performance optical design. In this configuration, the NAL consists of one planar surface and one aspherical surface with a curvature radius of R = − 23.26   mm (the negative sign indicates that the center of curvature lies to the left of the surface). For the optimized case, the aspherical surface of the NAL has an equivalent curvature radius of R = 23.26   mm and a conic coefficient of k = − 2886 . Based on these parameters, the corresponding input conditions of the laser beam and the nonlinear medium can be determined as follows: W 0 ≈ 0.5   mm , I 0 = 6.64   W / cm 2 , n 2 ≈ 10 − 6   cm 2 / W .

Figure 4.2D—footprint diagrams (left) and 3D—wavefront aberration (right) at different magnification. (a) M = 4; (b) M = 10; (c) M = 16; (d) M = 20.

The obtained value of the nonlinear refractive index coefficient n₂ falls within the typical range of organic dye materials, such as ORO, which exhibits a nonlinear coefficient of the same order of magnitude and a comparable linear refractive index [8]. Therefore, the use of organic dye materials (in the form of dry thin films) as the NAL in the Galilean system is entirely feasible. The selected nonlinear material exhibits relatively low absorption ( β ≈ 10 − 4 W / cm ) at the operating wavelength and a fast response time on the order of picoseconds. In addition, the use of a continuous wave laser helps reduce thermal effects under moderate power levels [21][27]. However, at high intensities, thermal accumulation and potential optical damage may occur, which should be considered negligible in practical implementations [26].

For comparison, a conventional Galilean beam expander reported in Ref. [9] exhibits a maximum RMS wavefront aberration of 0.1769λ. In contrast, the proposed system achieves significantly lower RMS values, with a maximum of 0.070λ across all configurations, indicating a substantial improvement in wavefront quality. Furthermore, unlike conventional designs that rely on precision-fabricated aspherical lenses, the nonlinear aspherical lens employed in this work does not require complex surface machining, thereby avoiding fabrication-induced errors and offering improved manufacturability while maintaining high optical performance.