Diffraction of Light | A-Level Edexcel 物理：光的衍射考点精讲

📚 Diffraction of Light | A-Level Edexcel 物理：光的衍射考点精讲

Light does not always travel in perfectly straight lines; when it encounters an obstacle or a narrow opening, it spreads out. This spreading, known as diffraction, is a fundamental wave property that provides direct evidence for the wave nature of light. For Edexcel A‑Level Physics, you need to understand the conditions for observable diffraction, the intensity patterns formed by single slits and diffraction gratings, and the mathematical relationships that allow us to calculate fringe positions, slit spacing, and the wavelength of light. Mastery of these ideas is essential for both the written examination and practical assessments.

光并不总是沿绝对直线传播；当它遇到障碍物或狭窄的开口时，会发生弯散。这种弯散现象叫做衍射，是光的波动性的直接证据。在 Edexcel A‑Level 物理中，你需要掌握可观察衍射的条件、单缝和衍射光栅形成的强度图样，以及用来计算条纹位置、光栅常数和光波长的数学关系。熟练掌握这些概念对笔试和实验考核都至关重要。

1. The Nature of Diffraction | 衍射的本质

Diffraction is the spreading of a wave as it passes through an aperture or around an obstacle. The amount of diffraction depends on the ratio of the wavelength λ to the size of the opening a. Significant diffraction occurs when the aperture width is comparable to the wavelength (a ≈ λ). If a is much larger than λ, the wave passes through with very little spreading, effectively travelling in straight lines. For light, wavelengths are extremely small (around 5 × 10⁻⁷ m), which explains why we do not normally notice light bending around everyday objects.

衍射是波通过孔隙或绕过障碍物时向外扩展的现象。衍射的明显程度取决于波长 λ 与开口尺寸 a 的比值。当开口宽度与波长相近时（a ≈ λ）会发生显著衍射。如果 a 远大于 λ，波通过时几乎不扩展，基本沿直线传播。由于光波波长极小（约 5 × 10⁻⁷ m），这就解释了为什么我们在日常生活中通常注意不到光绕过物体弯散。

Huygens’ principle provides a wave‑based explanation: every point on a wavefront acts as a source of secondary spherical wavelets. When part of the wavefront is blocked, these wavelets spread into the geometric shadow, producing diffraction effects. The principle elegantly accounts for both straight‑line propagation and diffraction under different conditions.

惠更斯原理提供了基于波动的解释：波前上的每一点都可以看作子波源，发出球面子波。当一部分波前被遮挡时，这些子波会扩展到几何阴影区，产生衍射效应。该原理巧妙地统一了直线传播与不同条件下的衍射现象。

2. Conditions for Observable Diffraction of Light | 光衍射可观察的条件

For a single slit of width a, the diffraction pattern becomes easily observable when a is of the order of 0.1 mm or less. In the laboratory, a laser is directed onto a narrow slit, and the pattern is projected onto a distant screen. A wider slit produces a narrow central maximum and fainter fringes, while a very narrow slit gives a broad central maximum. The condition for the first minimum in single‑slit diffraction is a sin θ = λ, where θ is the angle measured from the centre to the first dark fringe. This formula is a key result quoted on the Edexcel formula sheet.

对于宽度为 a 的单缝，当 a 约为 0.1 mm 或更小时，衍射图样易于观察。实验室中，将激光照射到一个窄缝上，图样投射在远处的屏幕上。较宽的缝产生较窄的中央明条纹和暗淡的次条纹，而非常窄的缝则产生宽阔的中央明条纹。单缝衍射第一暗纹的条件是 a sin θ = λ，其中 θ 是从中心到第一暗纹的夹角。该公式是 Edexcel 公式表上给出的关键结论。

Coherence of the light source is also important. Laser light is highly monochromatic and coherent, yielding sharp diffraction fringes. When using a white‑light source, the pattern becomes a spectrum, with each wavelength diffracted by a different angle, highlighting the dispersive nature of diffraction.

光源的相干性也很重要。激光高度单色且相干，能产生清晰的衍射条纹。使用白光光源时，图样变成彩色光谱，不同波长以不同角度衍射，体现出衍射的色散特性。

3. Single‑Slit Diffraction Pattern | 单缝衍射图样

When monochromatic light passes through a single narrow slit, the resulting pattern on a screen consists of a bright central maximum (fringe) that is roughly twice as wide as the subsidiary maxima on either side. The intensity falls off rapidly away from the centre. The central maximum is flanked by a series of dark fringes (minima) and much fainter bright fringes. The angular positions of the dark fringes are given by a sin θ = nλ, where n = 1, 2, 3, … and a is the slit width. Note that n = 0 corresponds to the central maximum, not a minimum.

单色光通过单一窄缝时，在屏幕上形成的图样包括一个明亮的中央极大（中央明条纹），其宽度大约是两侧次极大的两倍。强度随着远离中心迅速衰减。中央极大两侧分布着一系列暗纹（极小）和非常暗淡的明纹。暗纹的角位置满足 a sin θ = nλ，其中 n = 1, 2, 3, …，a 为缝宽。注意 n = 0 对应中央明纹，并非暗纹。

A common misconception is that the central maximum is twice as bright as the first side maximum; in reality, the central maximum carries most of the transmitted energy, and its intensity is far greater than any other fringe. The angular half‑width of the central maximum, from centre to first minimum, is Δθ ≈ λ/a (in radians), valid for small angles. This means narrower slits produce wider patterns, a direct inverse relationship.

常见误区是认为中央极大亮度是第一次极大的两倍；实际上中央极大承载了绝大部分透射能量，其强度远大于其他任何条纹。中央极大的角半宽，即中心到第一暗纹的角距离，在角度很小时约为 Δθ ≈ λ/a（弧度）。这表示缝越窄图样越宽，呈直接的反比关系。

4. Intensity Distribution for a Single Slit | 单缝的强度分布

The intensity I at an angle θ in single‑slit diffraction is described by a sinc‑squared function: I = I₀ [sin(β)/β]², where β = (πa sin θ)/λ. The central maximum occurs at β = 0 and gives I = I₀. Minima occur whenever sin β = 0 except β = 0, which leads to β = mπ, giving the condition a sin θ = mλ (m = ±1, ±2, …). Subsidiary maxima are located approximately midway between successive minima, but their intensities drop rapidly according to 1/(m+½)²π². For example, the first subsidiary maximum has only about 4.7% of the intensity of the central peak.

单缝衍射中，角度 θ 处的强度 I 满足 sinc 平方函数：I = I₀ [sin(β)/β]²，其中 β = (πa sin θ)/λ。中央极大位于 β = 0，强度为 I = I₀。暗纹出现在 sin β = 0 但 β ≠ 0 处，即 β = mπ，得出 a sin θ = mλ（m = ±1, ±2, …）。次极大约位于相邻暗纹的中间位置，但其强度按照 1/(m+½)²π² 快速下降。例如，第一次极大强度仅为中央峰强度的约 4.7%。

The Edexcel specification does not require you to derive this intensity formula, but understanding that the central maximum is bright and wide, while higher‑order maxima are faint, helps in sketching and interpreting patterns. The key diagram to remember shows a wide central peak with symmetrical, rapidly diminishing side peaks.

Edexcel 考纲不要求推导该强度公式，但理解中央极大明亮宽阔、而高阶极大黯淡的特点，有助于绘制和解读图样。需要记住的关键图示是一个宽阔的中央峰及其两侧迅速衰减的对称次峰。

5. Diffraction Grating and the Grating Equation | 衍射光栅与光栅方程

A diffraction grating consists of a large number of equally spaced parallel slits (or reflective grooves). The spacing between adjacent slits, d, is called the grating spacing. If a grating has N lines per metre, then d = 1/N. For a transmission grating, incident monochromatic light is split into a series of sharp, well‑separated maxima at angles given by the grating equation: d sin θ = nλ, where n is the order number (n = 0, 1, 2, …). The n = 0 order is the central bright line, n = 1 is the first order on either side, and so on.

衍射光栅由大量等间距平行狭缝（或反射凹槽）组成。相邻狭缝的间距 d 称为光栅常数。若光栅每米有 N 条刻线，则 d = 1/N。对于透射光栅，入射单色光被分成一系列尖锐、分得很开的明条纹，其角位置满足光栅方程：d sin θ = nλ，其中 n 是级数（n = 0, 1, 2, …）。n = 0 为零级中央明线，n = 1 为两侧的一级明线，以此类推。

The grating provides much sharper maxima than a double slit because the interference of many wavelets results in a very narrow bright fringe. The greater the number of slits illuminated, the brighter and sharper the maxima become. Spectrometers use diffraction gratings to separate light into its component wavelengths with high resolution, since different colours appear at different angles for the same order n.

光栅产生的明纹比双缝干涉尖锐得多，因为大量子波干涉的结果形成极窄的亮纹。被照亮的缝数越多，明纹越亮越锐利。光谱仪利用衍射光栅以高分辨率将光分解为不同波长成分，因为同一级 n 下不同颜色出现在不同角度。

6. Using the Grating Equation | 使用光栅方程

The grating equation d sin θ = nλ is central to many calculations. You may be asked to find an unknown wavelength from measured angles, or to determine how many orders are visible for a given grating and wavelength. The highest possible order occurs when sin θ ≤ 1, so n_max ≤ d/λ. For example, a grating with 300 lines per mm (d = 3.33 × 10⁻⁶ m) used with red light of wavelength 650 nm gives n_max = 3.33 × 10⁻⁶ / 650 × 10⁻⁹ ≈ 5.12, so the maximum visible order is n = 5.

光栅方程 d sin θ = nλ 是许多计算的核心。你可能需要根据测得的衍射角求未知波长，或计算给定光栅和波长下可见的级数。最高可能级数出现在 sin θ ≤ 1 时，因此 n_max ≤ d/λ。例如，每毫米 300 线的光栅（d = 3.33 × 10⁻⁶ m）与波长 650 nm 的红光，可得 n_max = 3.33 × 10⁻⁶ / 650 × 10⁻⁹ ≈ 5.12，故最大可见级数为 n = 5。

Be careful with units: d is usually given in metres or millimetres; convert all lengths to metres before substituting. Also, the angle θ is measured from the normal (the straight‑through direction). In many problems, you measure the angle between symmetrical orders and then halve it to find θ for the grating equation. Geometry links the angle to the distance from the central maximum on a screen at a known distance D: tan θ = x/D. For small angles, sin θ ≈ tan θ ≈ x/D, allowing the simple approximation λ ≈ xd / (nD).

注意单位：d 常以米或毫米给出；代入前须将所有长度单位化为米。此外，θ 从法线（直射方向）量起。许多题目中，你会测量对称级次之间的角度，然后取半得到用于光栅方程的 θ。几何关系将角度与屏幕上距离中央最大的距离 x 和已知距离 D 联系起来：tan θ = x/D。小角度时 sin θ ≈ tan θ ≈ x/D，从而可用近似式 λ ≈ xd / (nD) 简化计算。

7. Intensity Distribution for a Diffraction Grating | 衍射光栅的强度分布

The intensity pattern of a diffraction grating is the product of the single‑slit diffraction envelope and the multiple‑beam interference pattern. The result is a series of sharp principal maxima modulated by a broad single‑slit envelope. If the slit width a is comparable to the separation d, some orders can be missing if the position of an interference maximum coincides with a diffraction minimum from a single slit. Missing orders occur when d sin θ = nλ and a sin θ = mλ are satisfied simultaneously, which gives the condition d/a = n/m, an integer ratio.

衍射光栅的强度图样是单缝衍射包络与多光束干涉图样的乘积。结果是一系列锐利的主极大，被宽广的单缝包络所调制。如果缝宽 a 与间距 d 可以相比，当某干涉极大的位置恰好与单缝衍射极小重合时，会出现缺级现象。缺级发生的条件是同时满足 d sin θ = nλ 和 a sin θ = mλ，即 d/a = n/m，为整数比。

In practice, ruled gratings are blazed to concentrate most of the diffracted intensity into a particular order, enhancing brightness for that order. You may not be examined on blazing, but being aware of the interplay between the slit width and grating spacing helps explain why some diffraction orders appear much brighter than others.

实际中，闪耀光栅通过设计将大部分衍射强度集中到特定级次，增强该级的亮度。虽然考纲可能不涉及闪耀，但意识到缝宽与光栅常数之间的相互作用有助于解释为何某些衍射级次比其他级次亮得多。

8. Diffraction and the Wave Nature of Light | 衍射与光的波动性

Diffraction and interference phenomena provided pivotal historical evidence for the wave theory of light. Newton’s corpuscular theory could not satisfactorily explain why light bends around obstacles or produces fringes. Thomas Young’s double‑slit interference (1801) and later Fraunhofer’s diffraction experiments convincingly demonstrated that light undergoes constructive and destructive interference, a hallmark of wave behaviour. The ability to measure wavelengths of light using diffraction gratings further cemented the wave model, allowing precise determination of spectral lines.

衍射和干涉现象为光的波动说提供了关键的历史证据。牛顿的微粒说无法圆满解释光为何绕过障碍物或产生条纹。托马斯·杨的双缝干涉实验（1801 年）和后来的夫琅禾费衍射实验有力证明光会发生相长和相消干涉，这是波动行为的重要特征。利用衍射光栅测量光波波长进一步巩固了波动模型，使得精确测定光谱线成为可能。

However, the discovery of the photoelectric effect showed that light also exhibits particle‑like behaviour, leading to the modern concept of wave‑particle duality. In the Edexcel syllabus, diffraction is primarily treated as a wave phenomenon under the ‘Waves and Particle Nature of Light’ topic, but you should appreciate this dual character.

然而，光电效应的发现表明光也具有粒子性，从而产生了现代的波粒二象性概念。在 Edexcel 课程中，衍射主要作为“波与光的粒子性”主题下的波动现象来处理，但你需要理解这种双重属性。

9. Experimental Determination of Wavelength using a Grating | 利用光栅实验测定波长

A standard A‑Level practical involves using a laser and a diffraction grating to determine the wavelength of laser light. The grating is placed perpendicular to the beam, and the distances from the central spot to the first‑order spots on a screen are measured. Using the geometry tan θ = x/D, and for small angles approximating sin θ ≈ x/D, the wavelength is found from λ = xd / D (for n = 1). More accurate results are obtained by measuring the distance 2x between the two first‑order spots and taking the angle θ from arctan(x/D). Repeat readings and using a larger grating‑to‑screen distance D reduce percentage uncertainty.

一项标准的 A‑Level 实验是利用激光和衍射光栅测定激光波长。光栅垂直于光束放置，测量屏幕上中央点到一级亮点的距离。利用几何关系 tan θ = x/D，小角度时可用近似 sin θ ≈ x/D，对 n = 1 可得波长 λ = xd / D。若测量两一级亮点的间距 2x，再由 θ = arctan(x/D) 计算角度，结果更准确。重复读数并增大光栅到屏幕的距离 D 可以减小百分比不确定度。

Safety considerations are essential: lasers must be used with care to avoid direct eye exposure, and the beam should be terminated on a non‑reflective surface. The room may be darkened to improve fringe visibility. Although the grating equation itself has no inherent uncertainty, students often need to calculate percentage uncertainty in wavelength from uncertainties in x, D, and d.

安全注意事项必不可少：使用激光时必须避免直接照射眼睛，光束要终止在非反射表面。可以暗化房间以提高条纹可见度。虽然光栅方程本身没有固有不确定度，但学生常常需要根据 x、D 和 d 的不确定度计算波长的百分比不确定度。

10. Comparing Single‑Slit, Double‑Slit, and Grating Patterns | 单缝、双缝与光栅图样的比较

Understanding the differences between these patterns is a common exam requirement. A double‑slit produces equally spaced bright fringes (Young’s fringes) of nearly equal intensity within a single‑slit diffraction envelope. The fringe spacing Δy is given by Δy = λD/s, where s is the slit separation. In contrast, a single slit produces a broad central band and much fainter side fringes. A diffraction grating produces very sharp, well‑separated principal maxima, with large dark spaces between orders. The table below summarises key comparisons.

理解这些图样的区别是常见的考试要求。双缝产生等间距的明纹（杨氏条纹），强度近乎相等，但整体被单缝衍射包络所调制，条纹间距 Δy = λD/s，s 为双缝间距。单缝则产生宽阔的中央明带和暗淡的侧条纹。衍射光栅产生非常锐利、分得很开的主极大，各级次之间有大片暗区。下表总结了关键对比。

Feature 特征	Single Slit 单缝	Double Slit 双缝	Diffraction Grating 衍射光栅
Maxima spacing 明纹间距	Unequal, central wide 不等，中央宽	Equal within envelope 包络内等距	Equal in sin θ, sharp 按 sin θ 等距，锐利
Intensity fall‑off 强度衰减	Rapid from centre 从中心快速衰减	Gradual (envelope) 受单缝包络缓慢衰减	Slow within order; missing orders possible 级内缓慢；可能缺级
Formula 公式	a sin θ = nλ (minima) 暗纹	s sin θ = nλ (maxima) 明纹	d sin θ = nλ (maxima) 明纹
Number of sources 光源数	One extended source 单个扩展源	Two coherent sources 两个相干源	Many coherent sources 众多相干源

11. White Light Diffraction and Spectra | 白光衍射与光谱

When a diffraction grating is illuminated with white light, each order (except n = 0) spreads into a continuous spectrum, with violet deviated least and red deviated most, because the angle θ increases with wavelength for a given order. The central maximum (n = 0) remains white because all wavelengths combine at θ = 0. Overlapping of orders can occur: for instance, the violet end of the second‑order spectrum may overlap with the red end of the first‑order spectrum. This happens when (n+1)λ_violet ≈ n λ_red.

当白光照射衍射光栅时，除零级外，每一级都展成连续光谱，紫光偏转角最小，红光最大，这是因为对同一级次，θ 随波长增大而增大。中央极大（n = 0）仍为白色，因为所有波长在 θ = 0 处重合。可能出现级次重叠：例如，二级光谱的紫端可能与一级光谱的红端重叠。这发生在 (n+1)λ_紫 ≈ n λ_红时。

Such overlapping limits the useful range of a grating as a spectrometer; filters or order‑sorting techniques are employed to isolate a particular diffraction order. In the A‑Level context, you may be asked to identify the appearance of a white‑light fringe pattern and explain the colour sequence, linking back to the fact that sin θ ∝ λ.

这种重叠限制了光栅用作光谱仪的有效范围；实际中会使用滤光片或级次分离技术来隔离特定衍射级。在 A‑Level 范围内，你可能需要描述白光条纹图样的外观并解释颜色顺序，回到 sin θ ∝ λ 这一关系。

12. Key Pitfalls and Exam Tips | 常见失分点与应试技巧

Many students confuse the single‑slit equation a sin θ = nλ (for minima) with the double‑slit/grating equation (for maxima). Label them clearly. Always check whether a question asks for the angle to the first minimum or to the first bright fringe. Pay close attention to units: if d is given in lines per mm, convert to metres by taking the reciprocal and then dividing by 1000 or using 1 mm = 10⁻³ m. For a grating with N lines per mm, d = 1/(N × 10³) m. When calculating maximum order, take the integer part of d/λ, as partial orders do not appear.

许多学生将单缝公式 a sin θ = nλ（用于暗纹）与双缝/光栅公式（用于明纹）混淆。要清晰标注。务必核对题目是求到第一暗纹还是第一明纹的角度。特别注意单位：若 d 以每毫米线数给出，需要取其倒数并注意 1 mm = 10⁻³ m，即 d = 1/(N × 10³) m。计算最大级次时取 d/λ 的整数部分，因为非整数级不会出现。

In practical assessments, systematic errors such as misalignment of the grating, non‑normal incidence, or measuring the separation between the wrong pair of spots are common. Ensure the grating is perpendicular to the beam by checking that the zero‑order spot reflects straight back onto the laser aperture (with appropriate safety). Use a metre rule to measure D, and a Vernier calliper or travelling microscope to measure the distance between spots for improved precision.

在实验考核中，常见系统误差包括光栅未准直、非正入射，或测量了错误亮点对的间距。要确保光栅垂直于光束，可通过观察零级亮点是否反射回激光器孔径来检查（注意安全）。使用米尺测量 D，用游标卡尺或移测显微镜测量亮点间距以提高精度。

Published by TutorHao | A-Level Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Diffraction of Light | A-Level Edexcel 物理：光的衍射 考点精讲