Diffraction of Light | 光的衍射考点精讲

📚 Diffraction of Light | 光的衍射考点精讲

Mastering the topic of light diffraction is essential for both IB Physics and AQA Physics examinations. This article breaks down the core concepts, key formulas, and common pitfalls to help you achieve top marks in questions on diffraction through single slits, diffraction gratings, and resolving power. Whether you are analysing intensity patterns or calculating the minimum resolvable angle of a telescope, a clear understanding of wave behaviour at obstacles and apertures will set you apart.

掌握光的衍射对于 IB 物理和 AQA 物理考试都至关重要。本文将梳理核心概念、关键公式和常见错误，助你在单缝衍射、衍射光栅和分辨本领等题目中稳操胜券。无论你是在分析强度图样，还是计算望远镜的最小分辨角，对光波在障碍物与孔隙处行为的透彻理解都将成为你的得分利器。

1. What is Diffraction? | 什么是衍射？

Diffraction is the spreading out of waves as they pass through a narrow aperture or move around an obstacle. For light, this effect is most noticeable when the size of the aperture or obstacle is comparable to the wavelength of the light. Diffraction provides compelling evidence for the wave nature of light, and it explains why we can hear sound around corners but cannot see around them — sound wavelengths are much larger.

衍射是指波在通过狭窄缝隙或绕过障碍物时发生展延的现象。对光而言，当孔径或障碍物的尺寸与光的波长可比拟时，衍射效应最为明显。衍射是光具有波动性的有力证据，也解释了为什么我们能听到拐角处的声音却看不到——声波的波长要大得多。

In exam questions, you may be asked to predict how the fringe spacing changes when the slit width or wavelength is varied. A key idea is that the amount of diffraction increases as the aperture size decreases relative to the wavelength. Thus, if the slit width is narrowed, the central maximum becomes wider.

在考题中，你可能需要预测当缝宽或波长变化时条纹间距如何改变。一个核心思想是：相对于波长，孔径越小，衍射越显著。因此，如果缝宽变窄，中央明纹就会变宽。

2. Huygens-Fresnel Principle | 惠更斯-菲涅耳原理

The Huygens-Fresnel principle states that every point on a wavefront acts as a source of secondary spherical wavelets. The envelope of these wavelets at a later time constructs the new wavefront. When a wavefront encounters a slit, only those secondary sources within the slit contribute, causing the wave to spread into the geometrical shadow region.

惠更斯-菲涅耳原理指出，波前上的每一点都可以视为次级球面子波的波源。这些子波在之后某一时刻的包络面就构成了新的波前。当波前遇到一条狭缝时，只有狭缝内的次级波源起作用，使波进入几何阴影区域，产生展延。

This principle elegantly explains both the straight-line propagation of light in wide beams and the bending of light around obstacles when dimensions become small. In IB and AQA contexts you do not need to derive the principle mathematically, but you should be able to use it to justify the formation of diffraction fringes.

这一原理巧妙地解释了宽光束中光的直线传播，以及尺寸变小时光绕障碍物弯曲的现象。在 IB 和 AQA 的考查范围内，你不需要从数学上推导该原理，但应当能够用它来解释衍射条纹的形成。

3. Single-Slit Diffraction | 单缝衍射

When monochromatic light passes through a single slit of width a, a characteristic pattern of bright and dark fringes appears on a distant screen. The central maximum is wide and bright, flanked by much dimmer secondary maxima. This pattern arises from the interference of secondary wavelets originating from different points across the slit.

当单色光通过宽度为 a 的单缝时，远处的屏幕上会出现明暗相间的特征图样。中央明纹宽且亮，两侧的次级明纹则暗淡得多。这一图样源于从缝上各点发出的次级子波的干涉。

Destructive interference (dark fringes): a sinθ = mλ, m = ±1, ±2, ±3, …

相消干涉（暗纹）：a sinθ = mλ，m = ±1, ±2, ±3, …

Here θ is the angle measured from the centre of the pattern to the fringe. No simple formula exists for the exact positions of the bright secondary maxima. For small angles, the angular width of the central bright fringe is approximately 2λ / a (in radians), highlighting that narrowing the slit broadens the pattern.

式中 θ 是从图样中心到条纹的角位置。次级明纹并没有精确的简单公式。在小角度近似下，中央明纹的角宽度约为 2λ / a（弧度），这清晰地表明缝越窄，图样越宽。

4. Intensity Distribution in Single-Slit Diffraction | 单缝衍射的强度分布

The intensity I at an angle θ for a single slit is given by the expression:

单缝衍射在角度 θ 处的强度 I 由下式给出：

I = I₀ (sinβ / β)², where β = (π a sinθ) / λ

I = I₀ (sinβ / β)²，其中 β = (π a sinθ) / λ

I₀ is the intensity at the centre of the pattern. The intensity falls to zero whenever β = mπ, which directly recovers the dark-fringe condition a sinθ = mλ. The secondary maxima occur approximately halfway between successive minima, and their intensities drop rapidly: the first side maximum is only about 4.7% of I₀. Understanding this distribution helps explain why the central maximum dominates the pattern.

I₀ 是图样中心处的强度。每当 β = mπ 时，强度降至零，这直接回到了暗纹条件 a sinθ = mλ。次级明纹大约出现在相邻暗纹之间，且强度迅速衰减：第一侧明纹的强度仅为 I₀ 的约 4.7%。理解这一分布有助于解释为何中央明纹在衍射图样中占据主导。

While IB Higher Level may expect familiarity with this formula, AQA specifications often require qualitative understanding and the ability to sketch intensity against sinθ. Make sure you can label the central peak, minima, and secondary maxima clearly.

尽管 IB 高阶课程可能要求熟悉该公式，AQA 通常要求定性理解以及能够绘制强度随 sinθ 变化的草图。务必能清晰地标注中央峰、极小值和次级极大值。

5. Diffraction Gratings | 衍射光栅

A diffraction grating consists of many equally spaced slits (or rulings) that produce very sharp and well-separated bright fringes. The spacing between adjacent slits is called the grating element d. If there are N lines per metre, then d = 1 / N. Gratings are far more effective than a single slit for spectroscopy because they spread different wavelengths over larger angles and give brighter, narrower maxima.

衍射光栅由许多等间距的狭缝（或刻线）构成，能产生非常锐利且分隔清晰的明条纹。相邻狭缝间的距离称为光栅常数 d。如果每米有 N 条刻线，则 d = 1 / N。光栅在光谱学中比单缝有效得多，因为它能将不同波长以更大的角度散开，并给出更亮、更窄的极大值。

When answering exam questions on gratings, always convert lines per millimetre to d in metres. For example, a grating with 300 lines per mm has d = (1 × 10⁻³ m) / 300 = 3.33 × 10⁻⁶ m. This small size explains why precise alignment is needed to observe the diffraction orders.

在回答有关光栅的考题时，务必把每毫米的线数换算成以米为单位的 d。例如，每毫米 300 条线的光栅，d = (1 × 10⁻³ m) / 300 = 3.33 × 10⁻⁶ m。如此微小的尺寸也说明了为何需要精确对准才能观察到各级衍射。

6. The Grating Equation | 光栅方程

For a normally incident monochromatic beam, constructive interference occurs when the path difference between rays from adjacent slits equals an integer multiple of the wavelength. This leads to the grating equation:

对正入射的单色光束，当相邻狭缝光线之间的光程差等于波长的整数倍时，发生相长干涉，由此得到光栅方程：

d sinθ = nλ, n = 0, ±1, ±2, ±3, …

d sinθ = nλ，n = 0, ±1, ±2, ±3, …

Here n is the diffraction order. The zero order (n = 0) corresponds to θ = 0, where all wavelengths overlap to give a bright white fringe if white light is used. Higher orders appear symmetrically on either side. Since sinθ cannot exceed 1, the maximum observable order is limited to n_max ≤ d / λ.

式中 n 为衍射级次。零级 (n = 0) 对应 θ = 0，若使用白光则所有波长在此重叠，产生明亮的白色条纹。更高级次对称地分布在两侧。因为 sinθ 不能超过 1，可观测的最高级次受限于 n_max ≤ d / λ。

In practical problems, you may need to calculate the angular separation between two wavelengths in a given order or determine the number of visible orders. Always check whether your calculated sinθ is physically possible, and remember that the grating equation holds for transmission gratings as well as reflection gratings.

在应用题中，你可能需要计算给定级次中两种波长的角间距，或确定可见级次的数量。务必检验算出的 sinθ 是否物理可行，并记住光栅方程对透射光栅和反射光栅均适用。

7. Grating Spectra and Angular Dispersion | 光栅光谱与角色散

When white light passes through a grating, each wavelength produces maxima at slightly different angles, forming a continuous spectrum in each order (except n = 0). Violet light is deviated least, while red light is deviated most. This creates the familiar rainbow pattern that can be analysed to identify spectral lines of elements.

当白光通过光栅时，每一波长在略微不同的角度上产生极大值，从而在每一级次（除 n = 0 外）形成连续光谱。紫光偏转角最小，红光偏转角最大。这就形成了熟知的彩虹图样，可用于分析元素的谱线。

Angular dispersion describes how much the angle θ changes with wavelength for a given order. By differentiating d sinθ = nλ, we obtain (dθ / dλ) = n / (d cosθ). Greater dispersion is achieved by using gratings with smaller d (more lines per mm) or by observing higher orders. This is a favourite topic in AQA and IB data-analysis questions.

角色散描述在给定级次下角度 θ 随波长变化的程度。对 d sinθ = nλ 微分可得 (dθ / dλ) = n / (d cosθ)。使用 d 更小的光栅（每毫米更多线数）或观察更高级次，均可获得更大的色散。这是 AQA 和 IB 数据分析题中的热门话题。

8. Resolving Power and the Rayleigh Criterion | 分辨本领与瑞利判据

The Rayleigh criterion states that two point sources are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other. For a circular aperture of diameter D, the minimum angular separation θ_R (in radians) for resolution is:

瑞利判据指出，当一个衍射图样的中央极大与另一个的第一极小重合时，两个点源刚好能被分辨。对于直径为 D 的圆孔，可分辨的最小角间距 θ_R（以弧度计）为：

θ_R ≈ 1.22 λ / D

This formula applies to telescopes, microscopes, and even the human eye. The factor 1.22 arises from the first zero of the Bessel function describing the Airy disk. In exams, you should be able to explain why a larger objective lens or a shorter wavelength improves resolution.

该公式适用于望远镜、显微镜乃至人眼。因子 1.22 来源于描述艾里斑的贝塞尔函数的第一个零点。在考试中，你应能解释为什么更大的物镜口径或更短的波长能提高分辨本领。

For a diffraction grating, the chromatic resolving power R is defined as λ / Δλ = nN, where N is the total number of illuminated rulings. Thus, a grating with many lines can separate two very close wavelengths, making it a powerful tool in spectroscopy. Show your working clearly when calculating the minimum wavelength difference that can be resolved.

对于衍射光栅，色分辨本领 R 定义为 λ / Δλ = nN，其中 N 是被照射的刻线总数。因此，拥有大量刻线的光栅可以分开两条非常接近的波长，使其成为光谱学中的强大工具。在计算可分辨的最小波长差时，要清晰展示计算过程。

9. Double-Slit Interference with Diffraction Envelope | 双缝干涉与衍射包络

When a double-slit experiment is performed with slits of finite width, the interference fringes are modulated by a single-slit diffraction envelope. The bright interference fringes are strongest near the centre and fade away as the angle increases, eventually disappearing where the single-slit pattern has a minimum. This combined pattern is frequently tested in IB and AQA analysis tasks.

当双缝实验中的狭缝具有一定宽度时，干涉条纹会受到单缝衍射包络的调制。靠近中心的亮干涉条纹最强，随着角度增大逐渐衰减，并在单缝衍射极小处消失。这种组合图样在 IB 和 AQA 的分析题中常常出现。

Interference maxima: d sinθ = mλ, Diffraction minima: a sinθ = nλ

干涉极大：d sinθ = mλ，衍射极小：a sinθ = nλ

A ‘missing order’ occurs when a bright interference fringe coincides with a diffraction minimum, i.e., when d / a = m / n is a simple integer ratio. For example, if d = 2a, every second interference maximum is suppressed. Being able to sketch this envelope and identify missing orders demonstrates a deep understanding that examiners look for.

当某一干涉亮纹恰好与衍射极小重合时，即 d / a = m / n 成简单整数比，就会出现’缺级’现象。例如，若 d = 2a，则每隔一个干涉极大便被抑制。能够画出这种包络并识别缺级，展现的是考官所追寻的深层理解。

10. Applications of Diffraction | 衍射的应用

Diffraction is not merely an academic exercise; it underpins many technologies. Diffraction gratings are used in spectrometers to analyse the composition of stars, monitor pollutants, and even read the data from CDs and DVDs. X-ray diffraction (XRD) exploits the regular spacing of atoms in crystals to determine molecular structures, a technique central to chemistry and biology.

衍射并非纯粹的学术演练，它是许多技术的基石。衍射光栅用于光谱仪以分析恒星成分、监测污染物，甚至读取 CD 和 DVD 上的数据。X 射线衍射 (XRD) 利用晶体中原子规则排列的间距来确定分子结构，这一技术在化学和生物学中至关重要。

The Rayleigh criterion applies directly to the design of telescopes and microscopes. Radio telescopes, for instance, use very large dishes to achieve the angular resolution needed to separate distant cosmic sources. Electron microscopes exploit the tiny de Broglie wavelength of electrons to resolve details far smaller than optical microscopes can achieve. In an exam, linking a principle to a real-world application shows evaluative skill.

瑞利判据直接应用于望远镜和显微镜的设计。例如，射电望远镜使用巨大的碟形天线来达到分离遥远宇宙源所需的分辨率。电子显微镜则利用电子极小的德布罗意波长，去分辨比光学显微镜小得多的细节。在考试中，将原理与实际应用联系起来可以展示你的评估能力。

11. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students confuse the single-slit dark-fringe condition (a sinθ = mλ) with the grating constructive interference equation (d sinθ = nλ). Remember: a is the slit width for a single slit; d is the slit separation for a grating or double slit. Using the wrong symbol will cost marks even if the substitution is correct.

许多学生会把单缝暗纹条件 (a sinθ = mλ) 与光栅相长干涉方程 (d sinθ = nλ) 混淆。切记：a 是单缝的缝宽；d 是光栅或双缝的缝间距。用错符号即使代值正确也会扣分。

Other frequent errors include forgetting to convert angles to radians when using small-angle approximations, mixing up degrees and radians in dispersion calculations, and omitting the 1.22 factor in the Rayleigh criterion. Also, when asked to describe how a diffraction pattern changes with wavelength or slit width, make your answer comparative (e.g., ‘The fringes become wider’ rather than ‘The pattern changes’).

其他常见错误包括：使用小角度近似时忘记将角度转换为弧度，在色散计算中混淆度和弧度，以及遗漏瑞利判据中的 1.22 因子。此外，当被要求描述衍射图样如何随波长或缝宽变化时，回答应具有比较性（例如说’条纹变得更宽’而不是’图样会改变’）。

Finally, practise drawing diagrams: a single-slit intensity graph with zeroes labelled, a grating spectrum showing orders, and an Airy disk sketch for resolution. Clear, well-annotated diagrams can often earn full marks in qualitative questions. Always refer back to the fundamental wave model and check that your predictions are physically plausible.

最后，多练习绘图：带有零点标注的单缝强度图、显示级次的光栅光谱，以及用于分辨率的艾里斑草图。清晰且标注得当的图样常能在定性题中拿到满分。永远回归到基本的波动模型，并检验你的预测在物理上是否合理。

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Diffraction of Light | 光的衍射考点精讲