1. Bibliographic Information

• Title: Modeling light propagation for under-display sensing in a smartphone
• Authors: Qimeng Wang, Yi Liu, Guowei Zou, Zihao Liang, Haoteng Liu, Bo-Ru Yang, and Zong Qin.
• Affiliations: School of Electronics and Information Technology, Guangdong Province Key Laboratory of Display Material and Technology, State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University, Guangzhou, China.
• Journal/Conference: The paper is published by Optica Publishing Group. Optica (formerly OSA) is a highly reputable professional society for optics and photonics, and its publications are well-regarded in the field.
• Publication Year: The copyright notice indicates 2025. The supplement was published on July 14, 2025.
• Abstract: The paper addresses the challenge of designing under-display sensing (UDS) systems for bezel-less smartphones. The display panel distorts light, which degrades sensor performance. To solve this, the authors propose a universal model based on the Fresnel number to identify the correct light propagation regime for different sensors. They classify the under-display ambient light sensor (ALS), camera (UDC), and proximity sensor as operating in the geometric optics, Fraunhofer diffraction, and Fresnel diffraction regimes, respectively. Using a commercial smartphone, they validate their models through simulations and experiments, showing strong agreement for ALS angular response, UDC image blurring, and proximity sensor background noise. The model provides a rational basis for designing and optimizing UDS systems.
• Original Source Link: /files/papers/68e62e5303be06a1c585f06e/paper.pdf. The paper is published under the Optica Open Access Publishing Agreement.

2. Executive Summary

• Background & Motivation (Why):

  • The primary goal in modern smartphone design is to achieve a "bezel-less" or "infinity" display, which requires moving sensors like the front camera, ambient light sensor (ALS), and proximity sensor underneath the display panel. This technology is called Under-Display Sensing (UDS).
  • The core problem is that the display panel, with its complex, periodic pixel structure, significantly distorts the light passing through or reflecting off it. This distortion (e.g., obstruction, diffraction) degrades the accuracy and quality of the sensor signals.
  • Previous research treated each UDS system with a different optical model (e.g., geometric optics for one, wave optics for another) without a unified framework to determine which model to use. This ad-hoc approach makes systematic design and optimization difficult. The paper identifies a gap: the need for a universal criterion to select the appropriate light propagation model for any given UDS configuration.

• Main Contributions / Findings (What):

  1. A Universal Modeling Framework: The paper proposes using the Fresnel number as a single, fundamental criterion to classify the dominant light propagation mode for any UDS system. This unifies the analysis of different sensors under one theoretical umbrella.
  2. Classification and Modeling of Key UDS Systems: The authors apply this framework to three common sensors and demonstrate that:
     • Under-Display ALS operates in the geometric optics regime ($N \gg 1$), where performance is mainly limited by the physical obstruction of light by the pixel layout.
     • Under-Display Camera (UDC) operates in the Fraunhofer diffraction regime ($N \ll 1$), where the pixel layout acts as a diffraction grating, causing image blurring and artifacts.
     • Under-Display Proximity Sensor operates in the Fresnel diffraction regime ($N \approx 1$), a more complex near-field scenario. The paper provides an effective approximate model for this case.
  3. Experimental Validation on a Commercial Smartphone: The study's models are rigorously tested against experimental data obtained from a commercial smartphone (Vivo X70 Pro). The simulations for all three sensors show high accuracy and strong correlation with real-world measurements, validating the framework's effectiveness.
  4. Actionable Design Insights: The validated models provide practical guidance for optimizing UDS systems. For example, the study identifies an optimal 45-degree orientation for the proximity sensor's source and detector to minimize background noise.

3. Prerequisite Knowledge & Related Work

• Foundational Concepts:

  • Under-Display Sensing (UDS): The technology of placing sensors (camera, ALS, etc.) behind a smartphone's display panel to maximize the screen-to-body ratio.
  • Geometric Optics: A model of light propagation that describes light as rays traveling in straight lines. It is valid when the wavelength of light is much smaller than the objects it interacts with, and diffraction effects are negligible. This is also known as the "shadow regime."
  • Wave Optics: A model that describes light as waves. It is necessary when light interacts with objects comparable in size to its wavelength, leading to phenomena like diffraction and interference. Wave optics is further divided into two main regimes:
    • Fraunhofer Diffraction: Occurs in the "far-field," where both the light source and the observation screen are effectively at an infinite distance from the diffracting aperture. The diffraction pattern is the Fourier transform of the aperture.
    • Fresnel Diffraction: Occurs in the "near-field," where the source or observation screen (or both) are at a finite distance from the aperture. The calculations are more complex than in the Fraunhofer case.
  • Fresnel Number ($N$): A dimensionless parameter that determines whether a light propagation problem is in the near-field (Fresnel) or far-field (Fraunhofer) regime. It quantifies how much the wavefront curves across the aperture.

• Previous Works:

  • Prior studies on UDS tended to be siloed. Researchers modeling under-display cameras and transparent displays often used Fraunhofer diffraction to simulate image blurring and optimize pixel layouts to reduce diffraction artifacts [3, 20-25].
  • Work on under-display ALS and fingerprint sensors typically used geometric optics (ray tracing) to model the signal attenuation caused by physical obstruction from the Black Matrix (BM) and Pixel Define Layer (PDL) [7, 10].
  • However, geometric optics failed to explain phenomena in under-display proximity sensors, such as the orientation-dependent background noise. This indicated that a different physical model was needed.

• Differentiation:

  • The key innovation of this paper is not the invention of a new optical theory but the application of a unifying principle—the Fresnel number—to the specific, practical domain of smartphone UDS design.
  • Unlike previous works that applied a fixed model to a specific sensor, this paper provides a systematic method to first identify the correct physical regime and then apply the corresponding model. This framework is universal and can be applied to future UDS designs.

4. Methodology (Core Technology & Implementation)

The core of the paper's methodology is to use the Fresnel number to determine the correct optical model for a given UDS configuration.

• Principles: The Fresnel number ($N$) is calculated to classify the diffraction regime. The geometry for this calculation is shown in Figure 1.

  该图像为示意图，展示了光源发出的波前经过显示面板的衍射传播过程。光波由光源射出，经过半径为a的显示面板位置，产生圆形干涉条纹，最终到达接收器。图中标注了传播距离z和z'，体现了光的传播路径和衍射效应。

  The formula is: $$N = \frac { a ^ { 2 } } { \lambda } \left( \frac { 1 } { z } + \frac { 1 } { z ^ { \prime } } \right)$$ Where:

  • $N$: The dimensionless Fresnel number.

  • $a$: The characteristic radius of the aperture (e.g., the size of a transparent region in a pixel).

  • $\lambda$: The wavelength of the light.

  • $z$: The distance from the light source to the aperture (the display panel).

  • z': The distance from the aperture to the receiver.

  • Note: If lenses are present, $z$ and z' should be calculated using the positions of the source's or receiver's virtual images.

    The propagation regime is determined by the value of $N$:

  • If $N \gg 1$: Geometric Optics dominates. Light travels in straight rays, and shadows are sharp.

  • If $N \ll 1$: Fraunhofer Diffraction dominates. This is the far-field regime.

  • If $N \approx 1$: Fresnel Diffraction dominates. This is the near-field regime.

• Steps & Procedures for Each Sensor:

  1. Under-display Ambient Light Sensor (ALS)

  • Regime Identification: For the ALS setup in Figure 3, the detector is placed very close to the panel ($z' = H = 0.52 \text{ mm}$). The light source (ambient light) is far away ($z \to \infty$). With a pixel feature size $a$ of tens of microns and visible light $\lambda \approx 500 \text{ nm}$, the Fresnel number $N$ is calculated to be around 10. Since $N \gg 1$, the system is in the geometric optics regime.

    该图像为示意图，展示了显示面板与光传感器之间的光线路径。图中光线经过显示面板上可透光的“开口区域”后，传达到位于面板下方的光传感器，面板与传感器间距用H表示，光线呈斜向穿过开口区域。

  • Simulation Method: The model simulates the geometric obstruction of light.

    1. Obtain the pixel layout via microphotography and convert it to a binary image representing open (transparent) and blocked regions.

    2. For a given incident light angle $(\theta, \phi)$, project the sensor's area onto the bottom of the display panel to define a "receiving area."

    3. Project the open areas of the pixel layout through the panel's thickness $d$ along the same angle to define a "light transmitting area." Oblique rays are partially blocked by the side walls of the transparent "holes."

    4. The final received signal is proportional to the overlapping area, weighted by $\cos\theta$. For multi-layer panels, this process is repeated for each layer. The process is illustrated in Figure 4.

       

       2. Under-display Camera (UDC)

  • Regime Identification: In a UDC setup (Figure 6), the camera lens is crucial. The objects being photographed are typically far away ($z \to \infty$). The lens images these objects onto the sensor, meaning the sensor's virtual image as seen from the display panel is also at infinity ($z' \to \infty$). Therefore, both $1/z$ and $1/z'$ are zero, making $N \approx 0$. The system is deep in the Fraunhofer diffraction regime.

    该图像为示意图，展示了光线通过透镜聚焦到图像传感器的过程。屏幕发出的光线经过透镜折射后汇聚，最终在图像传感器上形成像。图中箭头表示光路方向，示意了透镜对屏幕光的聚焦作用。

  • Simulation Method:

    1. The image formed on the sensor is the convolution of the ideal, clear image with the camera's Point Spread Function (PSF).
    2. The PSF is the squared modulus of the Fourier transform of the aperture function. In this case, the aperture is the periodic pixel layout of the display.
    3. For a color image, the simulation is performed for separate RGB channels using representative wavelengths (e.g., 700 nm for Red, 540 nm for Green, 440 nm for Blue). Since the size of the diffraction pattern is proportional to wavelength, the PSFs for different colors will have different scales.
    4. Each channel of the reference image is convolved with its corresponding PSF, and the three resulting channels are recombined to form the final blurred, color-fringed image.

  3. Under-display Proximity Sensor

  • Regime Identification: The proximity sensor (Figure 8a) uses a nearby infrared (IR) source (Tx) and detector (Rx). The Tx is typically a weakly collimated LED. In the tested device, the virtual image of the IR source is at $z = 1.72 \text{ mm}$. The Rx detector is placed at the focal plane of its lens, so its virtual image is at infinity ($z' \to \infty$). With $\lambda = 1330 \text{ nm}$ and a pixel feature size $a$, the Fresnel number $N \approx 1$. This places the system in the Fresnel diffraction regime.

    该图像为多部分示意图和实验图。 (a)显示红外光源和探测器通过显示面板传播的光线及间隙g示意；(b)示意显示面板与光学元件关系及虚拟像位置；(c)左为显示面板微观结构照片，右为其二值化图像；(d)表示发射器(Tx)和接收器(Rx)的位置关系及最小串扰角度示意；(e)为不同角度θ下噪声强度的实验与模拟对比曲线；(f)为实际实验装置照片，标示显示面板、传感器及旋转机构。整体展示了光在显示面板与传感器间的传播模型和实验验证。

  • Simulation Method: Full Fresnel diffraction is complex. The authors use a known approximation for the setup in Figure 2, where a spherical wave is incident on an aperture, followed by a lens.

    该图像为示意图，展示了光波从屏幕经过透镜聚焦形成像的过程。图中标示了屏幕、透镜、焦平面和像平面的位置，光波经过屏幕到透镜的距离为d，屏幕到透镜的总距离为l，透镜到像平面的距离为l'，反映了光波传播与聚焦的几何关系。

    The resulting pattern on the detector is a geometrically scaled Fraunhofer diffraction pattern. The magnification $M$ depends on the distances. For the proximity sensor's reflection path (unfolded in Figure 8b), the magnification is given by $M = D_1 / (D_1 + D_2)$.

    1. Calculate the standard Fraunhofer diffraction pattern of the reflective pixel layout.
    2. Scale this pattern down by the magnification factor $M$. In the orientation experiment, $M=0.79$.
    3. The background noise is the amount of light from this scaled pattern that falls onto the receiver (Rx) area. By changing the relative orientation or gap, the amount of collected light changes.

5. Experimental Setup

• Datasets/Devices:

  • A commercial Vivo X70 Pro smartphone was the primary device for testing the under-display ALS and proximity sensor.
  • For the UDC, since a standalone commercial UDC was not available for modification, the authors used a Sony IMX800 camera and covered its lens with custom-fabricated amplitude masks that mimicked different pixel layouts. This setup is optically equivalent to a real UDC.
  • Microphotographs of the display panels (transmissive for ALS/UDC, reflective for proximity sensor) were taken to obtain the pixel layouts for simulations.

• Evaluation Metrics:

  • ALS: The Full Width at Half Maximum (FWHM) of the angular response curve was used to quantify the "viewing angle."
  • UDC: The Structural Similarity Index (SSIM) was used to quantitatively compare the similarity between simulated and experimentally captured images.
  • Proximity Sensor: The relative intensity of the background noise signal was measured and compared with simulation.

• Baselines:

  • ALS: The ideal cosinoidal law (Lambert's cosine law) for a bare sensor is used as a baseline to show the performance degradation.
  • UDC: The original, un-degraded reference image is the baseline for calculating SSIM and visually assessing quality loss.
  • Proximity Sensor: For the gap experiment, the simulation results are compared against the inverse-square law (expected from pure geometric optics) and a constant signal (expected from pure Fraunhofer diffraction) to prove that the system is in the Fresnel regime.

6. Results & Analysis

• Core Results:

  1. Under-display ALS (Figure 5)

  • The experiments confirm that the angular response of both tested ALS systems is significantly attenuated compared to the ideal cosinoidal curve.

  • The simulation results match the experimental data closely. For the Vivo X70 Pro, the experimental FWHM was 92.5°, and the simulation predicted 98.8° (an error of 6.3°). For the more complex COE display, the experimental FWHM was 60.9°, and the simulation predicted 69.9° (an error of 9°). This level of accuracy validates the geometric obstruction model.

    

    2. Under-display Camera (Figure 7)

  • The simulated images (Fig. 7d) successfully reproduce the characteristic blurring and color fringing artifacts seen in the experimental photographs (Fig. 7e).

  • The SSIM between simulated and experimental images is high (0.82 and 0.84 for the two layouts), confirming the model's accuracy.

  • The SSIM between the degraded experimental images and the original reference image is much lower (0.65 and 0.69), quantifying the image quality loss caused by the display panel.

    

    3. Under-display Proximity Sensor (Figures 8 & 9)

  • Orientation Analysis (Figure 8): The simulation predicted that background noise would be highest at 0° and 90° orientations and minimized at 45°. The experimental results show the exact same trend. This strong agreement validates the scaled Fraunhofer diffraction model and provides a clear design rule: orient the Tx-Rx pair at 45° to the pixel grid for minimal crosstalk.

  • Gap Analysis (Figure 9): The experiment shows that the background noise increases as the sensor-display gap increases. The Fresnel diffraction simulation correctly predicts this trend. This result directly contradicts geometric optics (which predicts a decrease via the inverse-square law) and Fraunhofer diffraction (which predicts a constant signal), providing conclusive evidence that the proximity sensor operates in the Fresnel regime.

    该图像包含一个图表和一组实验装置照片。图表（a）显示了不同传感器与显示屏间隙（mm）下噪声强度的模拟值、实验值及反平方定律的变化趋势对比。照片（b）展示了实验中的智能手机显示面板与红外接收器的实际装置及间隙放大图，明确标注了显示面板和红外接收器的位置及两者间的间隙。

• Supplemental Analysis: Rainbow Artifacts (Figure S1)

  • The paper's supplement analyzes the rainbow-like patterns seen when a strong light source reflects off a smartphone screen.

  • This phenomenon is identified as another case of Fraunhofer diffraction. The periodic pixel structure on the screen's surface acts as a reflection grating.

  • The simulation, based on the reflective microphotograph of the pixel layout, accurately reproduces the colorful, starburst-like diffraction pattern observed in a real photograph, further validating the model's applicability.

    该图像由四部分组成，属于插图和显微图。图(a)为示意图，展示光源发出光线透过智能手机屏幕并被人眼接收的过程；图(b)为智能手机屏幕像素的微观照片，标注了100微米比例尺；图(c)和图(d)分别为屏幕像素引起的光衍射图样，呈现彩色条纹和点阵状分布，反映不同光学传播效应。

7. Conclusion & Reflections

• Conclusion Summary: This study successfully develops and validates a universal model for analyzing light propagation in under-display sensing systems. By using the Fresnel number as a decisive criterion, the paper correctly identifies the distinct optical regimes governing under-display ALS (geometric optics), UDC (Fraunhofer diffraction), and proximity sensors (Fresnel diffraction). The proposed models are simple yet powerful, yielding simulation results that show strong agreement with experiments conducted on a commercial smartphone. This work provides a robust, physics-based framework for engineers to rationally design and optimize UDS technology.

• Limitations & Future Work:

  • The authors acknowledge that the UDC simulation has slight colorimetric inaccuracies. This is because the simplified RGB channel approach does not use the full spectral information of the light source and the camera sensor's quantum efficiency. A more physically accurate simulation would require hyperspectral data, which is often impractical to obtain.
  • There is a noticeable offset between the simulated and experimental curves in the proximity sensor gap experiment (Figure 9a). The authors attribute this to the practical difficulty of precisely measuring the tiny sensor-display gap in the experimental setup.
  • Future work could involve integrating these optical models with machine learning algorithms for image restoration (for UDC) or signal correction (for ALS/proximity sensors).

• Personal Insights & Critique:

  • The paper's primary strength is its elegance and practicality. It takes a complex engineering problem with many variables and boils it down to a single, fundamental physical principle (the Fresnel number). This approach is not only effective but also highly instructive.
  • The experimental validation is thorough and convincing. Using a commercial smartphone adds significant real-world relevance to the findings, bridging the gap between academic theory and industrial application.
  • The analysis of the proximity sensor is particularly insightful. It solves a puzzle that simpler models like geometric optics could not explain and provides a clear, actionable design recommendation (the 45° orientation) that directly improves performance.
  • This work is an excellent example of how first-principles thinking can lead to powerful and generalizable solutions in applied technology. The framework presented is not limited to the three sensors discussed; it can be readily applied to analyze future UDS technologies, such as under-display structured light sensors for face recognition or under-display LiDAR.