AS Physics: Diffraction of Light | AS 物理：光的衍射考点精讲

📚 AS Physics: Diffraction of Light | AS 物理：光的衍射考点精讲

Diffraction is a fundamental wave phenomenon in which light bends and spreads as it passes through a narrow aperture or around an obstacle. This topic is central to AS Physics, linking wave theory to the behaviour of light and underpinning key technologies such as spectroscopy. A thorough understanding of single-slit patterns, the diffraction grating equation, and the conditions for constructive and destructive interference is essential for examination success.

衍射是一种基本的波动现象，当光通过窄缝或绕过障碍物时会弯曲并扩散开来。这一主题是 AS 物理的核心内容，它把波动理论与光的行为联系起来，也是光谱学等关键技术的理论基础。透彻理解单缝图样、衍射光栅方程以及相长与相消干涉的条件，对于在考试中取得好成绩至关重要。

1. What is Diffraction? | 什么是衍射？

Diffraction is the spreading of waves as they pass through a gap or move around an obstacle. For light, the effect becomes significant when the size of the aperture or obstacle is comparable to the wavelength of the light. If the slit width is much larger than the wavelength, the wave passes straight through with minimal spreading; but as the slit narrows, the wavefront curves and the light fans out.

衍射是波通过缝隙或绕过障碍物时发生的扩散现象。对于光来说，当孔径或障碍物的尺寸与光波长可比拟时，衍射效应就会变得显著。如果缝宽远大于波长，波几乎不发生扩散而直线通过；但当缝变窄时，波前弯曲，光线向四周散开。

Diffraction provides strong evidence for the wave nature of light. It cannot be explained using a simple ray model, which predicts perfectly sharp shadows. The observed patterns of alternating bright and dark fringes confirm that light undergoes interference after being diffracted.

衍射为光的波动性提供了有力证据。它无法用简单的光线模型解释，因为光线模型预测的是清晰锐利的影子。观察到的明暗相间条纹证实，光在发生衍射后又经历了干涉。

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 position of the wavefront at a later time is the envelope that is tangent to all these wavelets. This principle elegantly explains how light spreads out after passing through a narrow slit: each point within the gap emits wavelets that construct the curved emerging wavefront.

惠更斯原理指出，波前上的每一点都可以看作是一系列次级球面子波的波源。随后时刻波前的新位置是所有子波的包络面。这一原理巧妙地解释了光通过窄缝后为什么会扩散开来：缝内每一点发出的子波共同构成了弯曲的出射波前。

When parallel light is incident on a slit, the wavefronts are planar. According to Huygens, the slit opening creates a set of point sources across its width. These sources are in phase and their wavelets overlap and interfere on a distant screen, producing a diffraction pattern.

当平行光入射到单缝时，波前是平面。根据惠更斯原理，缝的开口处产生了一排点波源。这些波源初相位相同，它们发出的子波在远处屏幕上重叠并发生干涉，从而形成了衍射图样。

3. Single-Slit Diffraction | 单缝衍射

When monochromatic light passes through a narrow single slit, a characteristic pattern is observed on a screen: a broad, intense central bright fringe flanked by a series of narrower, dimmer bright fringes on either side. The dark fringes are positions of complete destructive interference, while the bright fringes result from partial constructive interference.

当单色光通过一个窄单缝时，在屏幕上可以观察到一种特征图样：中央是一条又宽又亮的明条纹，两侧对称分布着若干较窄、较暗的明条纹。暗条纹处是完全相消干涉的位置，而明条纹处则发生了部分相长干涉。

The condition for a minimum (dark fringe) in single-slit diffraction is:

a sinθ = nλ (n = 1, 2, 3, …)

单缝衍射的极小（暗条纹）条件为：

a sinθ = nλ (n = 1, 2, 3, …)

Here a is the slit width, θ is the angle measured from the centre of the pattern to the minimum, λ is the wavelength, and n is an integer giving the order of the minimum. The central maximum lies between the first minima (n = 1) on either side.

这里 a 是缝宽，θ 是从图样中心到极小位置的夹角，λ 是波长，n 是一个整数，表示极小的级次。中央明纹位于两侧第一极小（n = 1）之间。

4. Intensity Distribution for a Single Slit | 单缝衍射的光强分布

The intensity of the bright fringes in a single-slit pattern decreases rapidly away from the centre. The central maximum contains the majority of the transmitted energy. The intensity I at an angle θ is given by:

I = I₀ [ sin(β) / β ]² where β = (πa sinθ)/λ

单缝图样中明条纹的光强随着远离中心而迅速减弱。中央明纹包含了绝大部分透射能量。在角度 θ 处的光强 I 由下式给出：

I = I₀ [ sin(β) / β ]² 其中 β = (πa sinθ)/λ

I₀ is the intensity at the centre of the pattern. When β = 0, the fraction sin(β)/β approaches 1, giving the central maximum. The secondary maxima occur approximately where sin(β) = 1, but their amplitudes are greatly reduced because of the 1/β² factor. In examination questions, you are not usually required to use this formula but must be able to sketch and label the intensity graph.

I₀ 是图样中心处的光强。当 β = 0 时，sin(β)/β 的极限趋于 1，对应于中央极大。次级极大大约出现在 sin(β) = 1 的位置，但由于 1/β² 因子，它们的振幅大幅降低。在考试题中，你通常不需要使用这个公式，但必须能够画出并标注光强分布图。

5. The Diffraction Grating | 衍射光栅

A diffraction grating consists of a large number of equally spaced, parallel slits or grooves on a transparent or reflective surface. Typical gratings used in school laboratories have 300, 600 or more lines per millimetre. The spacing d between adjacent slits is the reciprocal of the number of lines per unit length.

衍射光栅由大量等间距的平行狭缝或刻槽组成，刻制在透明或反射表面上。学校实验室常用的光栅每毫米有 300 条、600 条或更多刻线。相邻狭缝的间距 d 等于每单位长度刻线数的倒数。

When light passes through a transmission grating or reflects off a reflection grating, the many diffracted wavelets interfere. This produces a pattern of very sharp, intense principal maxima at well-defined angles, while the gaps between them are almost completely dark. Compared with the single slit, a grating gives much brighter and sharper maxima, making it ideal for precise wavelength measurements.

当光通过透射光栅或在反射光栅上反射时，许多衍射子波发生干涉。在确定的角度上会产生一系列非常锐利、强度很大的主极大，而它们之间的区域则几乎是全暗的。与单缝相比，光栅产生的明条纹更亮、更尖锐，因此特别适合用于精确测量波长。

6. The Grating Equation | 光栅方程

For a diffraction grating with light incident normally, constructive interference occurs when the path difference between waves from adjacent slits equals an integer number of wavelengths. This condition is summarised by the grating equation:

d sinθ = nλ (n = 0, 1, 2, 3, …)

对于垂直入射的光照射衍射光栅，当相邻狭缝发出的波之间的光程差等于波长的整数倍时，发生相长干涉。这一条件由光栅方程概括为：

d sinθ = nλ (n = 0, 1, 2, 3, …)

In this equation, d is the grating spacing (the distance between the centres of adjacent slits), θ is the angle of the diffracted beam measured from the normal, λ is the wavelength, and n is the order of the maximum. The zeroth order (n = 0) corresponds to the undeflected central beam, while first order (n = 1), second order (n = 2), and so on, appear symmetrically on both sides.

在此方程中，d 是光栅间距（相邻狭缝中心之间的距离），θ 是从法线量起的衍射角，λ 是波长，而 n 是极大的级次。零级 (n = 0) 对应于不发生偏折的中央光束，而一级 (n = 1)、二级 (n = 2) 等条纹对称地出现在两侧。

7. Orders of Maxima and Angular Dispersion | 极大级次与角色散

The highest observable order n_max is limited because sinθ cannot exceed 1. Therefore, n_max < d/λ. If d is comparable to λ, only a few orders appear; if d is much larger than λ, many orders may be visible, but the angular separation between them decreases.

可观察到的最高级次 n_max 受到 sinθ 不能超过 1 的限制。因此，n_max < d/λ。如果 d 与 λ 相近，只能出现少数几个级次；如果 d 远大于 λ，可能会出现很多个级次，但各级次之间的角间距会减小。

As the order n increases, the spread of wavelengths also increases, known as angular dispersion. In higher orders, a small range of wavelengths is spread over a larger angle, which helps to resolve closely spaced spectral lines. This is why diffraction gratings are so effective in spectrometers.

随着级次 n 的增大，不同波长的光的分散程度也增大，这称为角色散。在更高级次中，较小的波长范围会在更大的角度范围内展开，这有助于分辨靠得很近的光谱线。这就是衍射光栅在光谱仪中如此有效的原因。

8. White Light Diffraction Through a Grating | 白光通过光栅的衍射

When white light is incident on a diffraction grating, the central maximum (n = 0) remains white because all wavelengths overlap here without path difference. For n ≥ 1, each order produces a continuous spectrum, with violet light deviated least and red light deviated most, exactly the opposite of dispersion in a prism. The second order spectrum often overlaps with the first order spectrum at the violet end, and this overlap becomes more pronounced for higher orders.

当白光照到衍射光栅上时，中央极大（n = 0）仍为白色，因为在这个位置所有波长的光都没有光程差，直接重叠。当 n ≥ 1 时，每一级都会产生一个连续光谱，其中紫光偏折最小，红光偏折最大，这恰好与棱镜的色散相反。二级光谱在紫端常常会与一级光谱重叠，而且这种重叠在更高级次中更加明显。

To obtain a pure spectrum without overlapping, a single wavelength must be selected, or the first order alone must be used with careful measurement of angles. In exams, you may be asked to calculate the angular spread of the visible spectrum in a given order using the grating equation twice: once for red light (∼700 nm) and once for violet (∼400 nm).

要获得无重叠的纯净光谱，必须选择单一波长，或者仅使用一级光谱并仔细测量角度。在考试中，你可能需要利用光栅方程分别对红光（约 700 nm）和紫光（约 400 nm）进行计算，求出某个级次中可见光谱的角范围。

9. Comparison: Single Slit vs. Diffraction Grating | 单缝与衍射光栅的对比

Feature / 特征	Single Slit / 单缝	Diffraction Grating / 衍射光栅
Number of slits / 狭缝数量	1	Many (hundreds to thousands) / 许多（数百至数千）
Central maximum / 中央极大	Broad and very bright / 宽且很亮	Narrow and extremely bright / 窄且极亮
Position of maxima / 极大位置	Approximately halfway between minima / 大致在极小之间	Given by d sinθ = nλ / 由 d sinθ = nλ 给出
Minimum condition / 极小条件	a sinθ = nλ	Complex; minima occur between principal maxima / 复杂；极小出现在主极大之间
Sharpness of fringes / 条纹锐度	Broad with gradual fall-off / 较宽且光强缓慢下降	Very sharp, almost line-like / 非常尖锐，近乎线状
Application / 应用	Demonstrating wave nature / 演示波动性	Precise wavelength measurement, spectroscopy / 精确波长测量，光谱学

Note that the single-slit formula a sinθ = nλ gives minima, whereas the grating formula d sinθ = nλ gives maxima. A common exam mistake is to mix up the meanings of a and d, or to apply the wrong condition to the wrong setup.

请注意，单缝公式 a sinθ = nλ 给出的是极小位置，而光栅公式 d sinθ = nλ 给出的是极大位置。一个常见的考试错误是混淆 a 和 d 的含义，或者在错误的装置上套用了错误的条件。

10. Practical Applications of Diffraction | 衍射的实际应用

Diffraction gratings are widely used in spectrometers to analyse the spectral composition of light from stars, flames, or discharge tubes. By measuring the angles of the diffracted beams and applying d sinθ = nλ, scientists can identify elements through their characteristic emission or absorption spectra. This technique is fundamental in astrophysics and chemistry.

衍射光栅广泛用于光谱仪中，以分析来自恒星、火焰或放电管的光谱成分。通过测量衍射光束的角度并利用 d sinθ = nλ，科学家可以根据元素的特征发射或吸收光谱来识别元素。这一技术是天体物理学和化学中的基本方法。

In optical storage media such as CDs and DVDs, the closely spaced tracks act as a reflection grating, producing iridescent colours when white light falls on them. This everyday observation directly demonstrates diffraction and interference. Engineers also use the principles of single-slit diffraction to understand the resolution limits of optical instruments, such as telescopes and microscopes.

在 CD 和 DVD 等光学存储介质中，紧密排列的轨道起到反射光栅的作用，当白光照射时会呈现出虹彩般的颜色。这种日常生活中的观察直接展示了衍射和干涉现象。工程师们还利用单缝衍射的原理来理解望远镜和显微镜等光学仪器分辨率的极限。

11. Key Equations and Summary | 关键公式与总结

For your AS Physics examination, you must be fluent in the following equations and conditions:

在 AS 物理考试中，你必须熟练掌握以下方程和条件：

Single-slit minima: a sinθ = nλ

单缝极小： a sinθ = nλ
Diffraction grating maxima: d sinθ = nλ

衍射光栅极大： d sinθ = nλ
Grating spacing: d = 1 / N, where N is the number of lines per metre

光栅间距： d = 1 / N，其中 N 为每米的刻线数
Angular width of a fringe: often found by subtracting the angles for two adjacent minima or by doubling the angle to the first minimum (for central maximum width)

条纹的角宽度： 通常通过计算相邻两个极小的角度差来确定，或者把第一极小角度加倍（求中央明纹宽度）

Always remember that diffraction patterns are interference patterns from many coherent sources. The more slits there are, the sharper and brighter the maxima become. When solving problems, draw a clear diagram, label the normal, the path difference, and the angles, and convert all units to metres.

请始终记住，衍射图样是来自多个相干波源的干涉图样。狭缝越多，极大就越锐利、越明亮。在解题时，一定要画出清晰的示意图，标出法线、光程差和角度，并将所有单位转换为米。

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AS Physics: Diffraction of Light | AS 物理：光的衍射 考点精讲