Light and Geometric Optics光与几何光学

The wave model predicts diffraction, interference, and refraction — phenomena that require a spread-out oscillation to explain why light bends around corners, produces coloured fringes, and slows in dense media. The particle model is essential for the photoelectric effect (a single photon either ejects an electron or it does not; intensity doesn't matter, only frequency does) and for explaining why high-frequency radiation damages tissue more per quantum than low-frequency radiation.波动模型预测衍射、干涉与折射 — 这些现象需要弥散振荡来解释光绕角弯曲、产生彩色条纹、以及在致密介质中减速的原因。粒子模型对光电效应不可或缺（单个光子要么打出电子要么打不出；强度无关紧要，只有频率决定结果），也能解释为何高频辐射每个量子对组织的损伤比低频辐射更大。

Modern quantum mechanics unifies both views: light is described by a quantum field, and the wave-like or particle-like behaviour you observe depends on how you set up the experiment. For HS optics, the rule of thumb is: use the wave model for refraction, reflection, and diffraction (§1–§5 of this guide); use the photon model when energy transfer per quantum matters (photoelectric effect, photon energy questions).现代量子力学统一了两种观点：光由量子场描述，观测到的波动或粒子行为取决于实验的设置方式。在高中光学中，经验法则是：折射、反射与衍射用波动模型（本指南 §1–§5）；每个量子的能量转移有意义时（光电效应、光子能量问题）用光子模型。

The mirror does not actually swap left and right in space — it swaps front and back (depth). If you face north and look in a mirror, the reflection appears to face south. The "left-right reversal" you perceive is a cognitive effect: you mentally rotate the image around a vertical axis to face it, and that rotation swaps left and right. If you instead imagine rotating the image over a horizontal axis, it would look upside down with left and right unchanged. The mirror itself only reverses the dimension perpendicular to its surface.镜子并未真正在空间上交换左右，它交换的是前后（深度）。若你面朝北看镜子，映像看起来面朝南。你感知到的"左右互换"是认知效应：你在心理上将映像绕竖直轴旋转来面对它，而这个旋转就使左右互换了。若改为绕水平轴旋转映像，它看起来是上下颠倒但左右不变的。镜子本身只是反转了垂直于其镜面的那个维度。

Any two of these three rays locate the image graphically for a concave mirror (reverse the focal-point directions for a convex mirror):以下三条光线中任意两条可以图解法确定凹镜的像（凸镜则将焦点方向反转）：

1. A ray parallel to the principal axis reflects through the focal point $F$.平行于主轴的光线经反射后过焦点 $F$。
2. A ray through the focal point $F$ reflects parallel to the principal axis.过焦点 $F$ 的光线经反射后平行于主轴。
3. A ray directed at the centre of curvature $C$ reflects back along itself (it hits the mirror perpendicularly).指向曲率中心 $C$ 的光线沿原路反射回来（它垂直射到镜面）。

For a concave mirror with the object beyond $C$: image is real, inverted, diminished. Between $F$ and $C$: real, inverted, magnified. Inside $F$: virtual, upright, magnified (magnifying glass geometry). At $F$: no image (reflected rays are parallel). These cases are all predictable from the mirror equation; the ray diagram confirms the sign of $d_i$.物体在曲率中心 $C$ 之外时：实像、倒立、缩小。在 $F$ 与 $C$ 之间时：实像、倒立、放大。在 $F$ 之内时：虚像、正立、放大（放大镜几何）。在 $F$ 处：无像（反射光线平行）。这些情形均可由镜方程预测；光路图确认 $d_i$ 的符号。

In vacuum, the electromagnetic wave oscillates in free space at speed $c$. In a material, the oscillating electric field drives the bound electrons in atoms to oscillate, which re-emit the wave with a slight phase lag. The net effect of the superposition of the original wave and the re-emitted waves is a wave that travels at a lower phase velocity $v = c/n$. The frequency is unchanged (it is set by the source); only the wavelength shrinks: $\lambda_{\text{medium}} = \lambda_{\text{vacuum}}/n$. This is why a denser glass bends light more: its higher $n$ means a greater speed difference and a larger change in direction via Snell's law. Chromatic dispersion (the splitting of white light into colours by a prism) occurs because $n$ is slightly different for each wavelength, so each colour bends by a slightly different amount.在真空中，电磁波以速度 $c$ 在自由空间中振荡。在介质中，振荡的电场驱动原子中的束缚电子振荡，并以轻微的相位滞后重新辐射波。原波与重辐射波的叠加净效果是一列以较低相速度 $v = c/n$ 传播的波。频率不变（由光源决定）；只有波长缩短：$\lambda_{\text{介质}} = \lambda_{\text{真空}}/n$。这就是为何光密玻璃弯折光的程度更大：其较高的 $n$ 意味着更大的速度差，通过斯涅尔定律产生更大的方向改变。色散（棱镜将白光分解为颜色）的发生是因为 $n$ 对每种波长略有不同，故每种颜色弯折量略有差异。

A step-index fibre has a uniform-$n$ core surrounded by a lower-$n$ cladding. Light entering within the acceptance cone bounces down the fibre by TIR. The problem: different ray angles travel different path lengths, arriving at slightly different times — called modal dispersion — which limits the bandwidth of the cable.

A graded-index fibre has a core whose refractive index decreases smoothly from the centre outward. Rays that venture farther from the axis travel through lower-$n$ regions where they move faster, compensating for the longer path. All rays arrive at nearly the same time, dramatically increasing bandwidth. Graded-index multimode fibres are used in campus and data-centre networks; single-mode fibres (very thin core) eliminate modal dispersion entirely and are used for long-haul telecommunications.阶跃折射率光纤具有均匀折射率芯，外包较低折射率的包层。在接受锥内射入的光通过全内反射沿光纤传播。问题在于：不同角度的光线经过不同路径长度，到达时间略有不同 — 称为模式色散 — 限制了电缆带宽。

渐变折射率光纤的芯部折射率从中心向外平滑降低。偏离轴线较远的光线穿过折射率较低的区域，在那里速度较快，补偿了较长的路径。所有光线几乎同时到达，大幅提高带宽。渐变折射率多模光纤用于校园和数据中心网络；单模光纤（芯极细）完全消除模式色散，用于长途电信。

Optometrists specify lenses in dioptres (D), where power $P = 1/f$ with $f$ in metres. A converging lens of $f = 0.25$ m has $P = +4.0$ D; a diverging lens of $f = -0.5$ m has $P = -2.0$ D. When two thin lenses are in contact, their powers add: $P_{\text{total}} = P_1 + P_2$. This is why a pair of spectacle lenses is characterised by a single dioptre value.验光师以屈光度（D）标注透镜，其中光焦度 $P = 1/f$（$f$ 以米计）。$f = 0.25$ m 的会聚透镜光焦度为 $+4.0$ D；$f = -0.5$ m 的发散透镜为 $-2.0$ D。两块薄透镜紧贴时，光焦度相加：$P_{\text{总}} = P_1 + P_2$。这就是为何一副眼镜用单一屈光度值描述。

The lensmaker's equation relates focal length to the lens geometry and material: $\tfrac{1}{f} = (n-1)\!\left(\tfrac{1}{R_1} - \tfrac{1}{R_2}\right)$, where $n$ is the refractive index of the glass and $R_1$, $R_2$ are the signed radii of curvature of the two surfaces. This equation shows why a lens made of higher-$n$ glass (or with more curved surfaces) has a shorter focal length and greater bending power.磨镜者方程将焦距与透镜几何形状及材料相联系：$\tfrac{1}{f} = (n-1)\!\left(\tfrac{1}{R_1} - \tfrac{1}{R_2}\right)$，其中 $n$ 是玻璃折射率，$R_1$、$R_2$ 是两个面的带符号曲率半径。该方程说明为何由折射率较高的玻璃（或曲率更大的面）制成的透镜焦距更短、弯折能力更强。

A pinhole camera works with no lens at all: a tiny hole in one wall of a dark box projects an inverted image of the scene onto the opposite wall. Every point of the scene sends a thin cone of rays through the hole; a smaller hole gives a sharper (but dimmer) image. The image obeys the same geometry as a lens: the image-to-pinhole distance $v$ and the object-to-pinhole distance $u$ satisfy $h_i/h_o = v/u$ for image height $h_i$ and object height $h_o$.针孔相机无需任何透镜：暗箱一侧壁上的小孔将场景的倒立像投射到对侧壁上。场景的每一点通过小孔发射一细锥形光束；孔越小，像越清晰（但越暗）。像遵循与透镜相同的几何关系：像到针孔距离 $v$ 与物到针孔距离 $u$ 满足 $h_i/h_o = v/u$（$h_i$ 为像高，$h_o$ 为物高）。

A modern camera replaces the pinhole with a lens (or lens group). Focusing is achieved by moving the lens relative to the sensor, so that the lens equation $1/f = 1/d_o + 1/d_i$ is satisfied for the current object distance. Autofocus systems use a phase-detection sensor or a contrast-detection algorithm to determine the correct $d_i$ and drive the lens motor accordingly. The aperture (f-stop) controls depth of field: a small aperture (large f-number) keeps both near and far objects in focus; a large aperture gives a shallow depth of field and isolates subjects against a blurred background.现代相机用透镜（或透镜组）取代针孔。对焦通过移动透镜相对于传感器的位置来实现，使透镜方程 $1/f = 1/d_o + 1/d_i$ 对当前物距成立。自动对焦系统使用相位检测传感器或对比度检测算法确定正确的 $d_i$，并驱动镜头电机。光圈（f 值）控制景深：小光圈（大 f 值）使近处和远处物体都清晰；大光圈景深浅，使主体从模糊背景中突出。