Diffraction of Light for A-Level WJEC Physics | 光的衍射考点精讲

📚 Diffraction of Light for A-Level WJEC Physics | 光的衍射考点精讲

Diffraction is one of the most compelling pieces of evidence that light behaves as a wave. When light encounters an obstacle or passes through a narrow aperture with dimensions comparable to its wavelength, it spreads out into regions that would otherwise be in shadow according to geometrical optics. In the WJEC A-Level Physics specification, you are expected to describe and explain single-slit diffraction, diffraction gratings, and the way diffraction limits the resolving power of optical instruments. This revision guide covers all the key concepts, formulae, and typical exam questions for the diffraction of light.

衍射是光具有波动性的最有力证据之一。当光遇到尺寸与其波长相近的障碍物或窄缝时，它会扩展到根据几何光学本该是阴影的区域。在 WJEC A-Level 物理考试大纲中，你需要描述并解释单缝衍射、衍射光栅以及衍射如何限制光学仪器的分辨率。这篇考点精讲将覆盖光的衍射部分的所有关键概念、公式和常见考题。

1. What is Diffraction of Light? | 什么是光的衍射？

Diffraction is the bending and spreading of waves as they pass through a narrow gap or travel past an obstacle. For light, significant diffraction occurs only when the size of the aperture or obstacle is similar to the wavelength of the light (around 400-700 nm for visible light). This spreading into the geometric shadow is a property unique to waves and cannot be explained by ray optics.

衍射是波在穿过窄缝或经过障碍物时发生弯曲与扩展的现象。对于光，只有当孔径或障碍物的尺寸与光的波长（可见光大约在 400-700 nm）相近时，才会发生明显的衍射。这种向几何阴影区域的扩展是波所特有的性质，无法用光线光学解释。

In the context of WJEC exams, you might be asked to sketch diffraction patterns or to compare the amount of diffraction when the slit width is changed relative to the wavelength. A narrower slit or longer wavelength produces more spreading.

在 WJEC 考试中，你可能会被要求画出衍射图样，或者比较改变缝宽与波长相对大小时衍射程度的变化。缝越窄或波长越长，衍射扩展越显著。

2. Huygens’ Principle and Diffraction | 惠更斯原理与衍射

Christiaan Huygens proposed that every point on a wavefront acts as a source of secondary spherical wavelets. The envelope of these wavelets forms the new wavefront. When light passes through a single slit, each point within the slit acts as a secondary source. The superposition of these wavelets in different directions leads to the interference pattern we observe – a central bright fringe flanked by alternating dark and bright fringes.

克里斯蒂安·惠更斯提出，波前上的每一点都可以看作是一个次级球面波的点波源。这些次级波的包络面构成了新的波前。当光通过单缝时，缝内每一点都作为次级波源。这些次级波在不同方向上的叠加就形成了我们观察到的干涉图样——中央明条纹两侧交替出现暗纹和亮纹。

This principle beautifully links diffraction to interference. The single-slit diffraction pattern is essentially an interference pattern from an infinite number of coherent sources across the slit width. Understanding Huygens’ principle is fundamental to grasping why the central maximum is twice as wide as the subsidiary maxima.

惠更斯原理巧妙地将衍射与干涉联系了起来。单缝衍射图样本质上是来自缝宽方向上无数个相干波源所形成的干涉图样。理解惠更斯原理是掌握为何中央明纹宽度是其他次级明纹两倍的基础。

3. Single-Slit Diffraction Pattern | 单缝衍射图像

When monochromatic light passes through a single narrow slit, a characteristic pattern is observed on a distant screen: a bright, wide central maximum, with alternating dark and bright fringes on either side. The intensity of the subsidiary maxima decreases rapidly away from the centre. The central maximum is by far the brightest region.

当单色光通过单条窄缝时，在远处的屏幕上会观察到一个特征图样：宽阔明亮的中央明纹，两侧交替出现暗纹和亮纹。远离中心的次级明纹强度迅速减弱。中央明纹是所有区域中最亮的。

You must be able to describe how the pattern changes when the slit width a is varied. Decreasing a increases the amount of diffraction, widening the central maximum. Increasing the wavelength λ has the same effect. In an exam, you may need to sketch the intensity distribution and label the central peak and first-order minima.

你必须能够描述当缝宽 a 变化时图样如何改变。减小缝宽 a 会使衍射程度增大，中央明纹变宽。增大波长 λ 也有相同效果。考试中可能会要求你画出强度分布并标出中央明纹和一级暗纹。

4. Single-Slit Diffraction Formula and Dark Fringe Condition | 单缝衍射公式与暗纹条件

The positions of the dark fringes (minima) in a single-slit diffraction pattern are given by the equation:

单缝衍射图样中，暗纹（极小值）的位置由以下方程给出：

a sin θ = n λ

where a is the slit width, θ is the angle of the minimum relative to the straight-through direction, λ is the wavelength of the light, and n is a non-zero integer (n = ±1, ±2, ±3 …). Note that n = 0 corresponds to the centre of the central maximum, which is a bright fringe, not a dark fringe.

其中 a 为缝宽，θ 为暗纹相对于直线传播方向的偏角，λ 为光波长，n 为非零整数（n = ±1, ±2, ±3 …）。注意 n = 0 对应的是中央明纹的中心，它是一个亮纹，而不是暗纹。

From this condition, the angular half-width of the central maximum is the angle for n = 1: θ₁ = arcsin(λ/a). The central maximum therefore spans an angle of 2θ₁ between the first minima on either side. For small angles (θ in radians), sin θ ≈ θ, so θ₁ ≈ λ/a, giving a simple estimate of the central maximum width.

由该条件可得中央明纹的半角宽度即为 n=1 时的角度：θ₁ = arcsin(λ/a)。因此中央明纹的总角宽度为两侧一级暗纹之间的角度 2θ₁。对于小角度（θ 用弧度表示），sin θ ≈ θ，所以 θ₁ ≈ λ/a，这提供了中央明纹宽度的简便估算方法。

5. Diffraction Gratings | 衍射光栅

A diffraction grating consists of a large number of equally spaced parallel slits (or rulings). Gratings are usually specified by the number of lines per millimetre, N. The slit separation d is the distance between adjacent slits, given by d = 1/N (with appropriate unit conversion). A typical grating may have 300 lines per mm, giving d = 1/300 mm ≈ 3.33 × 10⁻⁶ m.

衍射光栅由大量等间距的平行狭缝（或刻线）组成。光栅通常以每毫米的线数 N 来标称。狭缝间距 d 是相邻缝之间的距离，即 d = 1/N（经过适当的单位换算）。典型的光栅可以是每毫米 300 线，则 d = 1/300 mm ≈ 3.33 × 10⁻⁶ m。

The diffraction grating produces an interference pattern with very sharp, well-defined bright maxima at specific angles. Because many slits contribute, the maxima are much brighter and narrower than in a double-slit experiment. This makes gratings ideal for measuring wavelengths precisely.

衍射光栅产生的干涉图样具有非常尖锐、清晰的亮纹（极大值），出现在特定的角度。由于是多缝参与干涉，这些明纹比双缝实验中的更明亮、更窄。这使得光栅非常适合精确测量波长。

6. The Grating Equation | 光栅方程

The angular positions of the principal maxima for a transmission grating are given by the grating equation:

透射光栅主极大的角位置由光栅方程给出：

d sin θ = n λ

where d is the slit separation, θ is the angle of the nth-order maximum measured from the normal, λ is the wavelength, and n is the order number (n = 0, 1, 2, 3 …). The central maximum, directly ahead, is the zero-order line (n = 0).

其中 d 为缝间距，θ 为第 n 级明纹与法线方向的夹角，λ 为波长，n 为光谱级数（n = 0, 1, 2, 3 …）。位于正前方的中央明纹是零级明纹（n = 0）。

To find the highest observable order, set sin θ = 1 (as sin θ ≤ 1). This gives n_max = floor(d/λ). For a grating with d = 2.0 × 10⁻⁶ m and λ = 600 nm = 6.0 × 10⁻⁷ m, n_max = 2.0 × 10⁻⁶ / 6.0 × 10⁻⁷ ≈ 3.33, so the maximum visible order is n = 3.

要找到可观察的最高级数，设 sin θ = 1（因为 sin θ ≤ 1）。得到 n_max = floor(d/λ)。例如光栅 d = 2.0 × 10⁻⁶ m，波长 λ = 600 nm = 6.0 × 10⁻⁷ m，n_max = 2.0 × 10⁻⁶ / 6.0 × 10⁻⁷ ≈ 3.33，因此可观察到的最高级数为 n = 3。

In WJEC problems, you may also be asked to find d from the number of lines per mm and then apply the grating equation, paying close attention to unit conversions (all lengths in metres).

在 WJEC 考题中，可能还会要求你由每毫米线数求 d，再应用光栅方程，并特别注意单位换算（所有长度单位用米）。

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

When white light is incident on a diffraction grating, each wavelength is diffracted to a slightly different angle for a given order, producing a continuous spectrum. The short-wavelength violet light is diffracted the least, while red light is diffracted the most. Thus in each non-zero order, a visible spectrum from violet to red appears.

当白光入射到衍射光栅上时，对于给定级数，各波长会被衍射到略微不同的角度，从而形成连续光谱。短波长的紫光衍射角度最小，红光衍射角度最大。因此在每一非零级中，都会出现从紫到红的可见光谱。

The angular dispersion D of a grating describes how much the angle changes with wavelength: D = Δθ/Δλ = n/(d cos θ). This shows that dispersion increases with order n and is larger for finer gratings (smaller d).

光栅的角色散 D 描述了角度随波长的变化率：D = Δθ/Δλ = n/(d cos θ)。这表明角色散随光谱级数 n 增大而增大，并且对于刻线更密的光栅（d 更小）也更大。

Overlapping of orders can occur when the second-order spectrum of a shorter wavelength overlaps with the third-order spectrum of a longer wavelength. This is an important consideration when analysing spectra with a grating.

当较短波长的二级光谱与较长波长的三级光谱发生重叠时，会出现级次重叠。这是用光栅分析光谱时需要重点考虑的因素。

8. Diffraction and Resolving Power: The Rayleigh Criterion | 衍射与分辨率：瑞利判据

Diffraction sets a fundamental limit on the ability of an optical instrument to distinguish two point sources that are close together. When light from a point source passes through a circular aperture (such as a telescope objective), it forms a diffraction pattern consisting of a central bright spot (the Airy disc) surrounded by rings.

衍射从根本上限制了光学仪器分辨两个接近点光源的能力。当点光源发出的光通过圆形孔径（如望远镜物镜）时，会形成一个由中央亮斑（艾里斑）和周围环组成的衍射图案。

The Rayleigh criterion states that two point sources are just resolvable 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 (in radians) is:

瑞利判据指出，当一幅衍射图样的中央极大恰好落在另一幅图样的第一极小上时，这两个点光源刚好可以分辨。对于直径为 D 的圆孔，最小分辨角 θ（弧度制）为：

θ ≈ 1.22 λ / D

This formula applies to telescopes, microscopes, and even the human eye. To improve resolution, you can use a larger aperture or observe with shorter wavelengths.

该公式适用于望远镜、显微镜，甚至人眼。要提高分辨率，可以使用更大的孔径或更短的波长。

In WJEC contexts, you should be able to apply the Rayleigh criterion to calculate the minimum resolvable angle for a telescope, and explain why radio telescopes need very large dishes due to the long wavelength of radio waves.

在 WJEC 体系中，你应能运用瑞利判据计算望远镜的最小分辨角，并解释由于无线电波波长较长，射电望远镜需要非常大的碟形天线的原因。

9. Applications of Diffraction of Light | 光的衍射的应用

Diffraction gratings are widely used in spectrometers to determine the wavelength of light from atomic emission or absorption spectra. They allow astronomers to identify elements in stars and chemists to analyse the composition of materials.

衍射光栅广泛应用于光谱仪中，用于从原子发射或吸收光谱中测定光的波长。它使天文学家能够识别恒星中的元素，化学家则可以分析物质的成分。

Diffraction also plays a key role in the design of compact discs and DVDs, where the closely spaced tracks act as a reflective grating, producing the rainbow-like colours. In microscopy, understanding diffraction is essential for pushing the limits of resolution using techniques such as fluorescence microscopy.

衍射在光盘和 DVD 的设计中也起着关键作用，其中密集排列的轨道充当了反射光栅，产生彩虹般的色彩。在显微术中，理解衍射对于利用荧光显微等技术突破分辨率极限至关重要。

Furthermore, the diffraction of light is exploited in holography and in advanced optical instruments. As a WJEC student, linking the theory of diffraction to such real-world applications can strengthen your written answers.

此外，光的衍射还被用于全息摄影和先进的光学仪器中。作为 WJEC 学生，将衍射理论与这些实际应用联系起来可以使你的书面回答更加出色。

10. Common Exam Points and Worked Examples | 常见考点总结与例题

Key points to remember:

需记住的关键点：

– Single-slit central maximum width increases when the slit becomes narrower or the wavelength increases.
– Grating maxima are described by d sin θ = nλ; for single-slit minima, use a sin θ = nλ (n non-zero).
– A grating with more lines per metre gives greater angular separation between orders and better spectral resolution.
– The highest order visible occurs when sin θ ≤ 1.
– Rayleigh criterion: θ = 1.22 λ/D for circular apertures.
– Unit conversion is critical: always convert mm or nm to metres before substituting into equations.

– 单缝的中央明纹宽度随缝变窄或波长变长而增大。
– 光栅明纹由 d sin θ = nλ 描述；单缝暗纹用 a sin θ = nλ （n 非零）。
– 每米刻线更多的光栅可使谱线间的角分离更大，光谱分辨率更好。
– 可观察到的最高级次由 sin θ ≤ 1 决定。
– 瑞利判据：对于圆孔，θ = 1.22 λ/D。
– 单位换算至关重要：代公式前务必将 mm 或 nm 转换为米。

Worked example:

例题：

A diffraction grating with 400 lines per mm is illuminated with monochromatic light of wavelength 500 nm. Calculate the angle of the second-order maximum and determine the maximum order visible.

用每毫米 400 线的衍射光栅照射波长为 500 nm 的单色光。计算二级明纹的角度，并求可观察到的最高级次。

Solution: d = 1×10⁻³ m / 400 = 2.5×10⁻⁶ m. For n = 2, sin θ = nλ/d = 2 × 500×10⁻⁹ / 2.5×10⁻⁶ = 1.0×10⁻⁶ / 2.5×10⁻⁶ = 0.40. So θ = arcsin(0.40) ≈ 23.6°. For maximum n, set sin θ = 1: n_max = d/λ = 2.5×10⁻⁶ / 500×10⁻⁹ = 5.0. Therefore the highest visible order is 5.

解答：d = 1×10⁻³ m / 400 = 2.5×10⁻⁶ m。对于 n = 2，sin θ = nλ/d = 2 × 500×10⁻⁹ / 2.5×10⁻⁶ = 1.0×10⁻⁶ / 2.5×10⁻⁶ = 0.40。所以 θ = arcsin(0.40) ≈ 23.6°。求最大 n 时，令 sin θ = 1：n_max = d/λ = 2.5×10⁻⁶ / 500×10⁻⁹ = 5.0。因此最高可见级次为 5。

Practising such calculations and ensuring you can describe the underlying wave principles will solidify your understanding of diffraction for the WJEC A-Level Physics examination.

练习此类计算并确保你能描述背后的波动原理，将巩固你对 WJEC A-Level 物理考试中衍射部分的理解。

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Diffraction of Light for A-Level WJEC Physics | 光的衍射考点精讲