| 摘要: | 在照明系統設計中，如何兼顧光能利用率、截止線銳利度與照度均勻性，一直是非成像光學的重要課題。傳統反射杯與單透鏡結構雖能塑造光型，但在截止線區域常見光能分布不均與最強點位置偏離的問題。本研究以倒三角形光強分布為核心，結合高斯權重分布建立模擬與分析流程，探討其經 柱狀透鏡陣列 (Cylindrical Lens Array, CLA) 調制後的光型行為。
在輸入場的建構上，研究定義倒三角形，並乘上高斯分布以模擬實際光場強度衰減。為了更貼近實際照明場景，進一步加入三種背景光比例，分別為1:2、1:3.5 與 1:5，以模擬高斯主峰與背景照度之間的能量分布差異，並觀察不同背景光強對截止線品質的影響。
模擬方法上，CLA 被視為線性位移不變(Linear Shift-Invariant, LSI)系統，其作用等效於輸入場與脈衝響應(Impulse Response)的卷積。本研究針對倒三角形頂角φ由90°至45°的變化進行模擬，並考慮不同脈衝響應寬度，觀察最強光點在遠場中相對截止線的位置與能量集中行為。結果顯示1:2 的背景光比例在截止線品質與最強點聚集效果上表現最佳，其餘比例則在均勻性或對比度上各有不足。這些結果不僅深化了對光場調制機制的理解，也為車燈設計、投影光學及非成像光學應用提供理論與設計參考。
;In lighting system design, achieving a balance among optical efficiency, cutoff sharpness, and illuminance uniformity has long been a critical issue in nonimaging optics. Conventional reflector cups and single-lens structures can shape the light distribution to some extent, but often result in uneven energy distribution near the cutoff region and displacement of the brightest point. This study focuses on an inverted triangular light-intensity pattern, combined with a Gaussian weighting distribution, to establish a simulation and analysis framework for investigating the optical behavior after modulation by a Cylindrical Lens Array (CLA).
For the construction of the input field, the inverted triangular pattern is defined and multiplied by a Gaussian function to simulate realistic light attenuation. To better approximate practical lighting conditions, three levels of background light compensation—1:2, 1:3.5, and 1:5—are introduced to model the energy balance between the Gaussian main peak and the background illumination, thereby evaluating the effect of background strength on cutoff quality.
In the simulation methodology, the CLA is modeled as a Linear Shift-Invariant (LSI) system, equivalent to the convolution of the input field with its Impulse Response. The study systematically examines inverted triangular apex angles ranging from 90° to 45°, under different impulse response widths, to analyze the position of the brightest point relative to the cutoff line and the associated energy redistribution in the far field. The results indicate that a 1:2 background light ratio provides the best performance in terms of cutoff sharpness and brightest point stability, while the other ratios show limitations in uniformity or contrast.
This work contributes in two aspects: first, by validating through numerical simulation the optical behavior of the inverted triangular pattern combined with a CLA, thereby filling a gap in existing studies on mask geometry and convolution effects; second, by demonstrating the critical role of background light compensation, and identifying the 1:2 ratio as the most effective configuration. These findings not only deepen the understanding of light-field modulation mechanisms but also provide theoretical and design references for automotive headlamps, projection optics, and other nonimaging optical applications. |
