📚 A2 Physics: Diffraction of Light | A2 物理：光的衍射 考点精讲
Diffraction is a fundamental wave phenomenon that reveals the wave nature of light. In A2 Physics, mastering diffraction means becoming confident with single-slit patterns, diffraction gratings, the relationships between wavelength, aperture size and fringe spacing, and the crucial concept of resolving power. This revision guide walks you through every key point, from Huygens’ principle to the Rayleigh criterion, with clear explanations and worked-through equations.
衍射是揭示光波动性的基本波动现象。在 A2 物理中，掌握衍射意味着要熟练掌握单缝图样、衍射光栅、波长与孔径尺寸及条纹间距的关系，以及至关重要的分辨本领概念。本考点精讲将带你梳理从惠更斯原理到瑞利判据的每一个关键点，配以清晰的讲解和详细推导的方程式。
1. What Is Diffraction? | 什么是衍射？
Diffraction is the bending and spreading of waves as they pass through a narrow opening or around an obstacle. The effect is most noticeable when the size of the aperture or obstacle is comparable to the wavelength of the wave. For light, this typically requires very small slits, since visible wavelengths are around 400-700 nm.
衍射是指波通过狭窄开口或遇到障碍物时发生弯曲和扩散的现象。当孔径或障碍物的尺寸与波的波长可比拟时，效果最为显著。对光而言，这通常需要极细的狭缝，因为可见光波长约在 400-700 纳米之间。
If the slit width is much larger than the wavelength, the wave passes through almost unchanged and diffraction is negligible. If the slit width is of the same order as the wavelength, the wave spreads out into a semicircular pattern.
如果缝宽远大于波长，波几乎不受影响地通过，衍射可以忽略。如果缝宽与波长数量级相同，波就会扩展成一个半圆形图样。
2. Huygens’ Principle | 惠更斯原理
Huygens’ principle states that every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront at a later time is the envelope that encloses all these wavelets. This model elegantly explains why diffraction occurs: when a wavefront is limited by a slit, the secondary sources at the edges produce wavelets that spread into the geometric shadow.
惠更斯原理指出，波前上的每一点都可以看作发出球面子波的次级波源。下一时刻的新波前就是包络这些子波的曲面。这个模型简洁地解释了衍射的成因：当波前被狭缝限制时，边缘的次级波源产生的子波会扩散到几何阴影区。
Using Huygens’ principle, the single-slit diffraction pattern can be understood as the superposition of an infinite number of point sources across the slit width.
利用惠更斯原理，单缝衍射图样可以理解为来自缝宽上无穷多个点光源的叠加。
3. Single-Slit Diffraction | 单缝衍射
When monochromatic light passes through a narrow single slit, a central bright fringe flanked by alternating dark and faint bright fringes is observed on a screen. The central maximum is twice as wide as the secondary maxima. The condition for destructive interference (dark fringes) is:
单色光通过一条狭窄的单缝时，在屏幕上会观察到中央亮纹，两侧交替排列着暗纹和较弱的亮纹。中央明纹的宽度是次级明纹的两倍。发生相消干涉（暗纹）的条件为：
a sinθ = nλ, n = ±1, ±2, ±3, …
where a is the slit width, θ is the angle from the centre to the dark fringe, λ is the wavelength, and n is an integer called the order of the minimum. Note there is no minimum for n=0; that direction corresponds to the central bright fringe.
其中 a 为缝宽，θ 为从中心到暗纹的角位置，λ 为波长，n 为整数阶数（暗纹级次）。注意 n=0 不存在暗纹，该方向对应中央亮纹。
For small angles, sinθ ≈ θ, and the linear position y on a screen at distance D from the slit is given by ydark = nλD / a. The width of the central bright fringe is therefore 2λD / a.
对于小角度，sinθ ≈ θ，屏幕上距狭缝距离为 D 的暗纹线位置由 ydark = nλD / a 给出。因此中央亮纹的宽度为 2λD / a。
4. Intensity Distribution in Single-Slit Diffraction | 单缝衍射的强度分布
The intensity of the single-slit diffraction pattern is not constant across the bright fringes. The intensity I at an angle θ is described by:
单缝衍射图样中的亮纹强度并不均匀。在角度 θ 处的强度 I 由下式描述：
I = I₀ (sinβ / β)², β = (π a sinθ) / λ
where I₀ is the intensity at the central maximum. The secondary maxima occur roughly when β = (2m+1)π/2, but their peak intensities drop rapidly: the first side maximum is only about 4.7% of the central intensity.
其中 I₀ 是中央极大的强度。次极大大致出现在 β = (2m+1)π/2 处，但其峰值强度迅速下降：第一侧极大的强度仅约为中央极大的 4.7%。
The dark fringes correspond to β = mπ (m ≠ 0), which leads exactly to a sinθ = mλ. This intensity formula confirms that most of the transmitted energy is concentrated in the central maximum.
暗纹对应 β = mπ (m ≠ 0)，正是 a sinθ = mλ 的条件。该强度公式证实大部分透射能量都集中在中央极大区域。
5. The Diffraction Grating | 衍射光栅
A diffraction grating consists of a large number of equally spaced parallel slits (or rulings). When monochromatic light is incident on a grating, constructive interference produces very sharp, intense maxima at angles given by the grating equation:
衍射光栅由大量等间距的平行狭缝（或刻线）组成。当单色光入射到光栅上时，相长干涉在满足光栅方程的角位置产生极锐利、明亮的极大值：
d sinθ = nλ, n = 0, ±1, ±2, …
where d is the distance between adjacent slits (the grating spacing), often given by d = 1/N with N being the number of lines per millimetre. The integer n is the order of diffraction.
其中 d 是相邻狭缝间距（光栅常数），通常由 d = 1/N 求得，N 为每毫米刻线数。整数 n 为衍射级次。
Because many slits contribute, the bright fringes are much sharper than those from a double slit, making the diffraction grating an excellent tool for measuring wavelength. The maximum possible order is limited by sinθ ≤ 1, so nmax ≤ d/λ.
由于大量狭缝参与贡献，亮条纹比双缝干涉的条纹锐利得多，这使得衍射光栅成为测量波长的绝佳工具。最大可能级次受限于 sinθ ≤ 1，因此 nmax ≤ d/λ。
6. Using the Grating Equation | 光栅方程的应用
To determine an unknown wavelength using a grating, you measure the diffraction angle θ for a known order n and grating spacing d. For example, if a grating with 300 lines per mm produces a first-order (n=1) maximum at an angle of 10.4°, the wavelength is:
要使用光栅测定未知波长，可在已知级次 n 和光栅常数 d 的情况下测量衍射角 θ。例如，每毫米 300 条刻线的光栅在 10.4° 处产生第一级（n=1）极大，则波长为：
λ = d sinθ / n = (1/300 mm) × sin 10.4° / 1 ≈ 6.0×10⁻⁴ mm = 600 nm
Gratings are also used to produce spectra. White light incident on a grating forms overlapping spectral orders; each order spreads the colours from violet (shortest wavelength) to red (longest wavelength), with greater dispersion at higher orders.
光栅还可用来产生光谱。白光入射到光栅上会形成重叠的光谱级次；每级都将颜色从紫（最短波长）到红（最长波长）散开，级次越高色散越大。
7. Combining Diffraction and Interference | 衍射与干涉的结合
In any real multi-slit system, such as a double slit or diffraction grating, the interference pattern is modulated by the diffraction envelope produced by a single slit. The overall intensity is the product of the single-slit diffraction term and the interference term. This explains why some bright fringes are missing if the path difference condition for a bright fringe coincides with the angle of a single-slit minimum.
在任何真实的多缝系统（如双缝或衍射光栅）中，干涉图样会受到单缝衍射包络的调制。总强度是单缝衍射项与干涉项的乘积。这就解释了当亮纹的光程差条件恰与单缝暗纹角度重合时为什么会出现缺级现象。
For a double slit with slit width a and separation d, the condition for missing orders is a sinθ = mλ (single-slit minimum) and d sinθ = nλ (interference maximum). Combining gives the missing orders when n = (d/a) m.
对于缝宽 a、缝距 d 的双缝，缺级条件为 a sinθ = mλ（单缝暗纹）同时 d sinθ = nλ（干涉极大）。联立可得缺级发生在 n = (d/a) m 时。
8. Resolving Power and the Rayleigh Criterion | 分辨本领与瑞利判据
The resolving power of an optical instrument describes its ability to distinguish two close point sources. According to the Rayleigh criterion, two sources are just resolved when the central maximum of one diffraction pattern falls exactly on the first minimum of the other.
光学仪器的分辨本领描述其区分两个相近点光源的能力。根据瑞利判据，当一个衍射图样的中央极大恰落在另一个衍射图样的第一暗纹上时，两个光源恰可分辨。
For a circular aperture (e.g., telescope), the minimum resolvable angular separation is:
对于圆孔径（如望远镜），最小可分辨角间距为：
θmin = 1.22 λ / D
where D is the diameter of the aperture. This shows that larger apertures and shorter wavelengths improve resolution.
其中 D 是孔径直径。这表明更大的孔径和更短的波长可改善分辨率。
9. Resolving Power of a Diffraction Grating | 衍射光栅的分辨本领
For a diffraction grating, the ability to separate two slightly different wavelengths in a given order is quantified by the resolving power R:
对于衍射光栅，在给定级次下区分两个波长略微不同的光的能力用量化的分辨本领 R 表示：
R = λ / Δλ = nN
where Δλ is the smallest resolvable wavelength difference, n is the order, and N is the total number of illuminated rulings. More rulings and higher orders yield greater resolving power.
其中 Δλ 为最小可分辨的波长差，n 为级次，N 为被照亮的刻线总数。刻线越多、级次越高，分辨本领越大。
This is why research-grade spectrometers use gratings with thousands of lines per millimetre and operate in high orders.
这就是为什么研究级光谱仪使用每毫米数千条刻线的光栅并在高级次下工作的原因。
10. Practical Applications of Diffraction | 衍射的实际应用
Diffraction underpins many modern technologies. CD and DVD discs act as reflective diffraction gratings; the tightly spaced tracks produce visible colour spectra. X-ray diffraction is used to determine crystal structures through Bragg’s law (which is also a diffraction effect). In astronomy, the diffraction limit of telescopes determines the smallest details visible on celestial bodies.
衍射是许多现代技术的基础。CD 和 DVD 光盘充当反射式衍射光栅；紧密排列的轨道产生可见的彩色光谱。X 射线衍射通过布拉格定律（也是一种衍射效应）用于确定晶体结构。在天文学中，望远镜的衍射极限决定了天体上可见的最细微结构。
Diffraction gratings are found in spectrometers for chemical analysis, and single-slit diffraction experiments in school laboratories demonstrate the fundamental wave properties of light.
衍射光栅用于化学分析的光谱仪中，而学校实验室中的单缝衍射实验则展示了光的基本波动性。
11. Quick Reference of Key Formulas | 关键公式速查
| Quantity | Formula |
|---|---|
| Single-slit minima | a sinθ = nλ, n = ±1, ±2,… |
| Central fringe width | 2λD / a |
| Grating maxima | d sinθ = nλ |
| Rayleigh criterion (circular aperture) | θmin = 1.22 λ / D |
| Grating resolving power | R = λ / Δλ = nN |
12. Exam Tips and Common Pitfalls | 考试技巧与常见易错点
Always convert all lengths to the same unit (often metres) before substituting into formulas. Remember that the central maximum of a single slit is twice the width of other maxima, a favourite exam fact. When using the grating equation, take care with the angle measured from the normal; sometimes the problem gives the angle between orders, which must be divided by two. For resolution questions, distinguish clearly between the Rayleigh criterion for circular apertures (factor 1.22) and the grating resolving power (no factor of 1.22).
代入公式前务必将所有长度换算成相同单位（常为米）。记住单缝的中央极大宽度是其他明纹的两倍，这是考试中爱考的知识点。使用光栅方程时，注意角度从法线量起；有时题目给出的是级次间的夹角，必须除以 2。关于分辨率的题目，要清楚区分圆孔径的瑞利判据（含因子 1.22）和光栅的分辨本领（不含因子 1.22）。
Draw clear, labelled diagrams showing wavefronts, ray paths and fringe patterns. For qualitative explanations, link back to Huygens’ principle and the superposition of wavelets.
画出清晰、带标注的示意图，标明波前、光线路径和条纹图样。进行定性解释时，要联系惠更斯原理和子波叠加。
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