Objective  The double free-form optical beam shaping system can adjust the spatial intensity distribution of the beam without altering its phase distribution. Still, it requires solving for the shape distribution of the double free-form optical surfaces by setting a virtual plane. This study reveals that applying the traditional single virtual surface method to design beam shaping systems with compact structures (short distance between double free-form optical elements) and large beam amplification (the ratio of the cutoff radius of the outgoing beam to the incident beam) ratios have some drawbacks, such as significant errors in solving for the double free-form optical surfaces and reduced shaping effectiveness, which generally includes the energy efficiency of the whole system for the beam and the irradiance uniformity of the beam after shaping.

  Methods  This paper presents a method for designing a free-form optical beam shaping system based on a virtual surface iteration strategy and the concept of misalignment is proposed to evaluate the difference between the obtained second free-form surface and the virtual surface. The first step involves the creation of a virtual plane at the vertex of the second free-form surface and the virtual surface serves as the target surface of the beam exiting from the first free-form surface. Subsequently, all the discrete points on the first free-form surface can be obtained by using the virtual surface and Snell's law, in this case, all discrete points on the second free-form surface can be obtained by using Snell's law given the outgoing beam of the first free-form surface and the target surface of the whole beam shaping system. Finally, an iterative process updates the virtual surface to approximate the true shape of the second free-form surface.

  Results and Discussions   The quantitative analysis examines the influence of the beam amplification ratio β and the axial distance D between the two optical elements on the misalignment between the virtual surface and the actual surface. A negative correlation is observed between β and misalignment, while a positive correlation exists between D and misalignment. Importantly, it is noticeable that with an increase in iterations, the value of misalignment rapidly approaches 0, thereby verifying the effectiveness of the Virtual Surface Iteration method. Two distinct beam shaping systems have been designed: a transmitted double free-form surfaces system and an off-axis two-mirror system. Simulation results demonstrate that both systems achieve over 95% irradiance uniformity (Tab.1) and more than 99% energy efficiency (Tab.1). Furthermore, employing the Single Virtual Surface method relatively enhances irradiance uniformity by 2.93% and energy efficiency by 8.930%.

  Conclusions  This paper presents a design method for the beam shaping system of the double free-form surface based on a virtual surface iteration strategy. The proposed method employs ray tracing to calculate discrete points on the free-form surface. It utilizes the virtual surface iteration strategy to minimize the misalignment between the virtual and real surfaces. This approach ensures that the virtual surface continuously approaches the actual shape of the second free-form surface, thereby enhancing the coupling between the free-form surfaces in the beam shaping system. Additionally, this study analyzes the correlation between misalignment and parameters of the beam shaping systems, concluding that the misalignment is positively associated with both the beam amplification ratio and the compactness of the spatial structure of the system. Subsequently, simulation software is employed to design and simulate a coaxial transmission beam shaping system as well as an off-axis double-mirror beam shaping system. These simulations yield outgoing beams with ideal irradiance uniformity and energy efficiency. Compared to virtual plane methods, our proposed approach significantly improves shaping effects, thus validating its effectiveness in laser processing, medical treatment, optical information processing, and other fields requiring laser beam shaping systems.