📚 Diffraction of Waves | 波的衍射
Diffraction is the spreading of waves as they pass through a narrow aperture or around an obstacle. It is a property common to all types of waves, including sound, water, and electromagnetic radiation. Diffraction occurs most noticeably when the size of the gap or obstacle is comparable to the wavelength of the incident wave, and its understanding is essential for explaining phenomena ranging from the audible sound around corners to the detailed spectra produced by a diffraction grating in optical physics.
衍射是波在穿过狭窄缝隙或绕过障碍物时发生的展宽现象。它是所有类型的波（包括声波、水波和电磁波）的共同性质。当缝隙或障碍物的尺寸与入射波的波长可相比拟时，衍射现象最为显著。理解衍射对于解释从拐角处可听到的声音到光学物理中由衍射光栅产生的精细光谱等一系列现象至关重要。
1. Introduction to Diffraction | 衍射简介
Diffraction refers to the bending and spreading of a wave as it encounters an edge, a slit, or a small obstacle. Instead of continuing to travel only in straight lines, the wavefronts curve and spread out into the geometrical shadow region. This behaviour cannot be explained by simple ray optics; it is a direct consequence of the wave nature of light and other wave phenomena.
衍射是指波在遇到边缘、缝隙或小型障碍物时发生弯曲和扩展的现象。波前不会只沿直线传播，而是会弯曲并扩展到几何阴影区域。这种行为无法用简单的射线光学来解释；它是光以及其他波动现象的波动本质的直接结果。
n depends on the ratio of the wavelength λ to the size of the aperture or obstacle a. If λ ≪ a or λ > a, the spreading is significant and a clear diffraction pattern can be observed.
衍射的程度取决于波长 λ 与缝隙或障碍物尺寸 a 的比值。若 λ 远小于 a，衍射效应很弱，波基本上沿直线传播，并产生锐利的阴影。当 λ 与 a 相近或 λ 大于 a 时，波的展宽十分显著，可观察到清晰的衍射图样。
2. Huygens' Principle and the Mechanism of Diffraction | 惠更斯原理与衍射机制
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 is tangent to all these secondary wavelets. When a wavefront passes through a narrow slit, only a small number of point sources remain active, and the wavelets spread out in many directions, giving rise to the characteristic diffraction pattern.
惠更斯原理指出，波前上的每一点都可视为次球面子波的波源。经过一段时间后的新波前是所有次子波的包络面。当波前穿过狭窄缝隙时，只有少数点源保持活跃，子波向各个方向扩展，从而形成特有的衍射图样。
In single-slit diffraction, these secondary wavelets interfere with each other. The superposition of wavelets at different angles leads to points of constructive and destructive interference, forming a pattern of bright and dark fringes on a screen placed far away. This link between diffraction and interference is fundamental: diffraction patterns are essentially interference patterns produced by the continuous distribution of secondary sources across the slit.
在单缝衍射中，这些次子波会彼此干涉。不同角度上的子波叠加导致某些位置出现相长干涉和相消干涉，从而在远处的屏幕上形成明暗条纹的图样。衍射与干涉之间的这种联系是基础性的：衍射图样本质上是由缝隙上连续分布的次波源产生的干涉图样。
3. Conditions for Significant Diffraction | 显著衍射的条件
For a wave to diffract appreciably, the size of the gap or obstacle must be of the same order of magnitude as the wavelength λ of the wave. If the gap width b is much larger than λ, the wave passes through with only a small amount of spreading at the edges and a sharp central beam is observed. Conversely, if b is comparable to or smaller than λ, the wave spreads out almost uniformly in all directions beyond the slit.
为了使波发生明显的衍射，缝隙或障碍物的尺寸必须与波的波长 λ 处于同一数量级。如果缝隙宽度 b 远大于 λ，波通过时仅在边缘有少量扩展，可观察到锐利的中央光束。相反，若 b 与 λ 可相比拟或更小，波在缝隙后几乎均匀地向各个方向展宽。
This explains why sound waves, with wavelengths ranging from a few centimetres to several metres, can easily diffract around everyday objects such as doors and furniture, while visible light, with wavelengths around 400–700 nm, requires very narrow slits or gratings to show diffraction effects. In the context of the A-level syllabus, typical slit widths for light diffraction are on the order of tens to hundreds of micrometres.
这解释了为何波长从几厘米到几米不等的声波很容易绕过门和家具等日常物体发生衍射，而波长在 400–700 nm 左右的可见光则需要非常窄的缝隙或光栅才能展现衍射效应。在 A-level 考纲范围内，光衍射使用的典型缝宽为几十到几百微米。
4. Single-Slit Diffraction: Central Maximum and Minima | 单缝衍射：中央极大与暗纹
When monochromatic light passes through a single narrow slit of width b, a diffraction pattern is observed on a distant screen. The pattern consists of a broad, bright central maximum flanked by much dimmer secondary maxima. The central maximum is twice as wide as the secondary maxima and contains the vast majority of the transmitted light intensity.
当单色光通过宽度为 b 的单缝时，在远处屏幕上可观察到衍射图样。该图样由一个宽阔明亮的中央极大以及两侧暗得多的次级极大组成。中央极大的宽度是次级极大的两倍，并且包含了绝大部分的透射光强。
The positions of the dark fringes (minima) in single-slit diffraction are given by the condition:
b sin θ = nλ, n = 1, 2, 3, ...
th of the light. The central maximum is bounded by the first minima at angles where sin θ ≈ λ / b. For small angles, the angular width of the central maximum is approximately 2λ / b.
单缝衍射中暗纹（极小）的位置由下列条件给出：
b sin θ = nλ, n = 1, 2, 3, ...
其中，θ 为入射光方向与屏幕上极小位置之间的夹角，b 为缝宽，λ 为光的波长。中央极大被 n = 1 的第一暗纹所包围，其角度满足 sin θ ≈ λ / b。对于小角度，中央极大的角宽度近似为 2λ / b。
ble-slit interference pattern a similar-looking equation d sin θ = nλ gives the positions of bright fringes. Students must be careful to distinguish these cases.
需要特别注意的是，在单缝衍射中 b sin θ = nλ 的条件给出的是相消干涉（暗纹），而在双缝干涉图样中，一个形式上相似的方程 d sin θ = nλ 给出的是亮纹的位置。学生必须小心区分这两种情形。
5. Intensity Distribution in Single-Slit Diffraction | 单缝衍射的强度分布
The intensity of the bright fringes falls off rapidly away from the centre. The central maximum is the brightest region, and the intensity of the secondary maxima decreases significantly: the first-order secondary maximum on each side typically has only about 4.7% of the central intensity, and the second-order maximum is even weaker. The intensity distribution can be plotted against sin θ or against position on the screen.
亮纹的强度随着远离中心而迅速衰减。中央极大是最亮的区域，次级极大的强度大幅降低：两侧的一级次级极大通常仅为中央强度的约 4.7%，二级极大则更弱。可将强度分布绘制为随 sin θ 或屏幕上位置变化的曲线。
A graph of intensity versus angle shows a strong central peak, with minima at integer multiples of λ / b (in terms of sin θ). Between these minima are low secondary peaks. When the slit is made narrower, the central maximum spreads wider, and when the slit is widened, the central maximum becomes narrower. This inverse relationship between slit width and fringe spacing is a key characteristic of single-slit diffraction.
强度对角度的关系图显示出一个很强的中央峰，并在 sin θ 为 λ / b 的整数倍处出现极小值。在这些极小值之间是较低的次级峰。当缝宽变窄时，中央极大展得更宽；当缝宽增宽时，中央极大变窄。这种缝宽与条纹间距之间的反比关系是单缝衍射的关键特征。
6. Diffraction Grating: Multiple Slits and Sharp Maxima | 衍射光栅：多缝与锐利极大
A diffraction grating consists of a large number of equally spaced, identical parallel slits. When monochromatic light falls on the grating, each slit acts as a source of diffracted wavelets. The waves from all the slits interfere to produce an interference pattern that is characterised by very sharp, well-defined bright fringes known as principal maxima.
衍射光栅由大量等间距、完全相同的平行狭缝构成。当单色光照射光栅时，每条狭缝都成为衍射子波的波源。所有狭缝发出的波相互干涉，产生一种干涉图样，其特点是具有非常尖锐、清晰可辨的亮纹，称为主极大。
The sharpness of the maxima increases with the number of slits N. For a grating with thousands of lines per centimetre, the bright fringes become narrow spectral lines, making the grating an excellent tool for measuring wavelengths and for producing spectra. Between the principal maxima there are many subsidiary maxima that effectively form a dark background.
极大值的锐度随着狭缝数量 N 的增加而提高。对于每厘米有数千条线的光栅，亮纹会变为极窄的光谱线，这使得光栅成为测量波长和产生光谱的极佳工具。在主极大之间还存在着许多次极大，它们实际上形成了暗背景。
7. The Grating Equation and Orders of Maxima | 光栅方程与光谱级数
For a diffraction grating, the directions of the principal maxima are given by the grating equation:
d sin θ = nλ, n = 0, 1, 2, 3, ...
r n is called the order of the maximum: n = 0 gives the central zero-order maximum, n = 1 gives the first-order maximum on either side, and so on.
对于衍射光栅，主极大的方向由光栅方程给出：
d sin θ = nλ, n = 0, 1, 2, 3, ...
其中 d 是相邻狭缝的间距（光栅常数），若已知每单位长度条数 N，则常表示为 d = 1 / N。整数 n 称为光谱级数：n = 0 给出中央零级极大，n = 1 给出两侧的一级极大，依此类推。
han the slit width b. Because d is typically much larger than the wavelength, many orders can be observed. The maximum possible order is found by setting sin θ = 1, giving nmax ≤ d / λ. Any order for which d / λ falls below n is not visible.
与单缝不同，光栅的亮纹在 sin θ 上是等间距的，并且条件中涉及的是光栅常数 d 而非缝宽 b。由于 d 通常远大于波长，因此可以观察到多个级数。可能的最大级数可通过令 sin θ = 1 得到，即 nmax ≤ d / λ。若 d / λ 小于 n，则无法观察到该级条纹。
8. White Light and Diffraction Spectra | 白光与衍射光谱
If a diffraction grating is illuminated by a source of white light, each wavelength component produces its own set of maxima at angles determined by the grating equation. The shorter wavelengths (blue/violet) are diffracted through smaller angles, while the longer wavelengths (red) are deviated more. As a result, the central zero-order maximum remains white, but each higher order produces a continuous spectrum.
若用白光照射衍射光栅，则每一种波长成分都会根据光栅方程在其各自的角度上产生极大值。较短波长（蓝/紫）的衍射角较小，而较长波长（红）的偏折更大。因此，中央零级极大仍然是白色的，但每一较高级次都会产生连续光谱。
In the first-order spectrum, violet appears closest to the centre and red appears furthest. For higher orders, the spectra become more spread out, and overlapping can occur: the blue end of the second order may overlap with the red end of the third order, depending on the grating spacing. This is why gratings are often used in spectrometers to separate and analyse different wavelengths.
在一级光谱中，紫色最靠近中心，红色最远。对于更高级次，光谱会变得更加分散，且可能出现重叠：第二级的蓝端可能与第三级的红端重叠，具体取决于光栅常数。这正是光栅常用于光谱仪中以分离和分析不同波长的原因。
9. Diffraction of Sound and Radio Waves | 声波与无线电波的衍射
Diffraction is not limited to light; it is a universal wave phenomenon. Sound waves in air have wavelengths from about 17 m (for 20 Hz) to a few centimetres (for high frequencies). Because typical doorways and windows are comparable in size to these wavelengths, sound can diffract significantly around obstacles. This allows a person to hear a conversation from a neighbouring room even when the door is only slightly open.
衍射并不局限于光；它是一种普遍的波动现象。空气中的声波波长范围从约 17 m（20 Hz）到几厘米（高频）。由于典型的门、窗尺寸与这些波长可相比拟，声音能明显地绕障碍物衍射。这使得人们即使只开着一条门缝，也能听到隔壁房间的谈话。
Similarly, radio waves with long wavelengths (AM radio) can diffract around hills and buildings, reaching receivers that are not in the direct line of sight. In contrast, short-wavelength waves such as microwaves and visible light show much less diffraction around large obstacles and require clear paths or repeaters for transmission. Understanding this dependence on wavelength is vital for designing communication systems.
同样，长波长的无线电波（如调幅广播）可以绕射过山丘和建筑物，到达不在直视路径上的接收器。相比之下，微波和可见光等短波长波在大型障碍物周围的衍射要弱得多，传输时通常需要通畅路径或中继器。理解这种对波长的依赖对于设计通信系统至关重要。
10. Comparison of Diffraction and Interference | 衍射与干涉的比较
Although both terms are often used together, diffraction and interference describe slightly different aspects of wave superposition. Diffraction is the bending and spreading of waves caused by the presence of an obstacle or slit, while interference refers to the superposition of two or more coherent waves from separate sources. Nevertheless, every diffraction pattern is essentially an interference pattern arising from the secondary wavelets across an aperture.
尽管这两个术语经常一起使用，衍射和干涉描述的是波叠加的略有不同的方面。衍射是由障碍物或缝隙的存在而引起的波的弯曲和扩展，而干涉则指来自不同波源的两个或多个相干波的叠加。然而，每一衍射图样本质上都是由孔径上的次子波产生的干涉图样。
e), the maxima are given by d sin θ = nλ; and for a diffraction grating, the sharp principal maxima also follow d sin θ = nλ. The single-slit pattern has a wide central maximum, while the double-slit pattern has equally spaced fringes of nearly equal brightness when the single-slit envelope is neglected. The grating produces extremely narrow, bright lines.
在考试中，正确应用公式很重要。对单缝，极小值方程为 b sin θ = nλ；对双缝（干涉），极大值由 d sin θ = nλ 给出；对衍射光栅，锐利的主极大也遵循 d sin θ = nλ。单缝图样具有宽的中央极大，而双缝图样在不考虑单缝包络时，呈现等间距且亮度几乎相等的条纹。光栅则产生极窄的明线。
11. Measuring Wavelength with a Diffraction Grating | 使用衍射光栅测量波长
A standard A-level experiment involves determining the wavelength of a laser using a diffraction grating. The grating with a known number of lines per millimetre (e.g., 300 lines mm⁻¹) is placed in a holder, and a laser beam is directed perpendicularly through it. The diffracted spots are observed on a screen at a measured distance L from the grating. The distance x between the central spot and the first-order spot is recorded.
A-level 中一个标准实验是利用衍射光栅测定激光的波长。将已知每毫米线数（例如 300 lines mm⁻¹）的光栅安装在支架上，让激光束垂直穿过。在距离光栅已知距离 L 处的屏幕上可观察到衍射光点。记录中央光点与一级光点之间的距离 x。
nλ with n = 1, the wavelength λ is found. If possible, the measurement is repeated for the second order to improve accuracy. Systematic errors, such as non-perpendicular incidence and measurement of distances, should be discussed.
角度 θ 可由 tan θ = x / L 获得。小角度下 sin θ ≈ tan θ，但更精确的做法是计算出 sin θ。利用光栅方程 d sin θ = nλ，取 n = 1，即可求出波长 λ。若条件允许，可重复测量二级条纹以提高准确度。应讨论系统误差，如非垂直入射以及距离测量等。
12. Summary and Key Points | 总结与要点
ure size is comparable to the wavelength. Single-slit diffraction produces a distinct pattern with a broad central maximum governed by b sin θ = nλ for minima. A diffraction grating, with its many slits, generates very sharp principal maxima given by d sin θ = nλ, allowing precise measurement of wavelength.
总而言之，衍射是波遇到小型缝隙或障碍物时发生的扩展现象。仅在缝隙尺寸与波长可相比拟时，衍射效应才会显著。单缝衍射产生独特的图样，其宽大的中央极大由极小值公式 b sin θ = nλ 决定。衍射光栅具有大量狭缝，可生成由 d sin θ = nλ 给出的非常锐利的主极大，从而实现对波长的精确测量。
All waves, including sound, water, and electromagnetic radiation, exhibit diffraction. The practical applications range from the design of optical instruments, where diffraction limits resolution, to the use of reflection gratings in compact discs and the analysis of crystal structures through X‑ray diffraction. A solid understanding of this topic is essential for success in CIE A‑Level Physics and for appreciating the wave nature of light and sound.
所有波，包括声波、水波和电磁辐射，都会表现出衍射现象。其实际应用范围甚广，从光学仪器设计中衍射对分辨率的限制，到光盘中反射光栅的应用，再到通过 X 射线衍射分析晶体结构。扎实地理解这一主题对于在 CIE A-Level 物理中取得成功以及理解光和声音的波动本性都至关重要。
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