Light Diffraction Phenomena and Common Question Types | 光的衍射现象与常见题型

📚 Light Diffraction Phenomena and Common Question Types | 光的衍射现象与常见题型

Diffraction is one of the most compelling pieces of evidence that light behaves as a wave. When light passes through a narrow slit or around an obstacle, it bends and spreads out, producing patterns of bright and dark fringes. Understanding diffraction is essential not only for wave optics theory but also for solving a wide range of exam problems in A-level and international curricula.

衍射是证明光具有波动性的最重要现象之一。当光通过狭缝或绕过障碍物时，会发生弯曲和扩展，形成明暗相间的条纹图案。理解衍射不仅是波动光学理论的核心，也是解决A-level及国际课程中大量物理考题的关键。

1. What Is Diffraction? | 什么是衍射？

Diffraction is the spreading or bending of waves as they pass through an aperture or around an obstacle. The effect is most noticeable when the size of the slit or obstacle is comparable to the wavelength of the wave. For light, this means apertures on the order of micrometres to millimetres produce observable diffraction patterns.

衍射是波在通过孔缝或绕过障碍物时发生的扩展或弯曲现象。当缝或障碍物的尺寸与波的波长相近时，这一效应最为显著。对于光而言，微米到毫米量级的缝宽即可产生可观察的衍射图样。

The diffraction pattern of a single narrow slit consists of a broad, intense central maximum flanked by narrower, less intense secondary maxima, with dark minima in between. The positions of these minima depend on the slit width and the wavelength of light.

单缝衍射图样由一个宽而明亮的中央亮纹和两侧较窄、亮度较低的次级亮纹组成，其间由暗纹隔开。这些暗纹的位置取决于缝宽和光的波长。

2. Conditions for Significant Diffraction | 产生明显衍射的条件

For diffraction to be easily observed, the obstacle or slit dimension a must be comparable to the wavelength λ. In practice, a should be within a few orders of magnitude of λ. If the slit is much wider than the wavelength, diffraction is negligible and light travels in straight lines (geometrical optics).

要容易地观察到衍射现象，障碍物或缝的尺寸 a 必须与波长 λ 相当。实际上，a 应处于 λ 的若干个数量级范围之内。若缝远宽于波长，衍射可忽略，光近似沿直线传播（几何光学）。

Visible light: λ ≈ 400–700 nm; slit width for diffraction ≈ 0.1–1 mm is suitable.
Sound: λ ≈ 0.1–1 m; diffraction around doorways is common.
X-rays: λ ≈ 0.1 nm; diffraction requires atomic-scale periodic structures (crystals).

可见光：λ ≈ 400–700 nm；缝宽约为 0.1–1 mm 时可观察到衍射。
声波：λ ≈ 0.1–1 m；绕过门口发生衍射很常见。
X射线：λ ≈ 0.1 nm；衍射需要原子尺度的周期性结构（晶体）。

3. Single-Slit Diffraction and Minima Positions | 单缝衍射与暗纹位置

Consider a single slit of width a illuminated by monochromatic light of wavelength λ. The condition for destructive interference (dark minima) at an angle θ is given by:

a sin θ = nλ, n = ±1, ±2, ±3, …

Here n is the order of the minimum. There is no minimum for n = 0 because that corresponds to the central maximum. The first minimum occurs when a sin θ = λ, which defines the edge of the central bright fringe.

设缝宽为 a，用波长为 λ 的单色光照射单缝。暗纹（相消干涉）条件为：

a sin θ = nλ，n = ±1，±2，±3，…

其中 n 为暗纹级数。n = 0 对应中央亮纹，因此没有暗纹。第一级暗纹出现在 a sin θ = λ 处，它确定了中央亮纹的边缘。

For small angles (θ in radians), sin θ ≈ tan θ ≈ θ. If the screen is at distance L from the slit, the linear distance y of the nth minimum from the centre is:

y = L tan θ ≈ L sin θ = (nλL)/a

对于小角度（θ 以弧度为单位），sin θ ≈ tan θ ≈ θ。若屏幕距缝为 L，则第 n 级暗纹距中心的线距离为：

y = L tan θ ≈ L sin θ = (nλL)/a

4. Width of the Central Maximum | 中央亮纹的宽度

The central maximum is the bright region between the first minima on either side. Its angular width is the angle subtended by these two minima:

Δθ = 2θ₁ = 2 sin⁻¹(λ/a) ≈ 2λ/a (for small θ)

The linear width on a screen at distance L is:

Δy = 2y₁ = 2L tan θ₁ ≈ 2λL/a

中央亮纹是两侧第一级暗纹之间的亮区。其角宽度为这两个暗纹所对应的张角：

Δθ = 2θ₁ = 2 sin⁻¹(λ/a) ≈ 2λ/a（小角度近似）

在距离为 L 的屏幕上的线宽度为：

Δy = 2y₁ = 2L tan θ₁ ≈ 2λL/a

Key consequences: as slit width a decreases, the diffraction pattern spreads out more (wider central maximum). As wavelength λ increases, the pattern also spreads. This is opposite to what happens in geometrical shadows.

重要结论：缝宽 a 减小时，衍射图样更扩展（中央亮纹变宽）。波长 λ 增大时，图样同样变宽。这与几何阴影的情况正好相反。

5. Diffraction by a Circular Aperture and Resolution | 圆孔衍射与分辨本领

When light passes through a circular aperture of diameter D, the diffraction pattern consists of a central Airy disc surrounded by faint rings. The angular radius of the first dark ring is:

θ = 1.22 λ / D

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. Thus the minimum angular separation for resolution is:

θ_min = 1.22 λ / D

当光通过直径为 D 的圆孔时，衍射图样由中央艾里斑和周围模糊的圆环组成。第一暗环的角半径为：

θ = 1.22 λ / D

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，两个点光源恰好可分辨。因此，可分辨的最小角间距为：

θ_min = 1.22 λ / D

Quantity	Symbol / Formula	Note
Single-slit minima	a sin θ = nλ	n = ±1, ±2, …
Central max width (small θ)	Δy ≈ 2λL/a	Between first minima
Circular aperture resolution	θ_min = 1.22λ/D	Rayleigh criterion

物理量	符号/公式	说明
单缝暗纹	a sin θ = nλ	n = ±1，±2，…
中央亮纹宽度（小角度）	Δy ≈ 2λL/a	两第一级暗纹之间
圆孔衍射分辨极限	θ_min = 1.22λ/D	瑞利判据

6. Diffraction vs. Interference | 衍射与干涉的区别

Both diffraction and interference arise from the superposition of waves, but they are not the same phenomenon. Interference refers to the superposition of waves from two or more coherent sources (e.g., double slits). Diffraction refers to the spreading of waves from a single source or aperture due to Huygens’ principle, where each point in the aperture acts as a new source.

衍射和干涉都源于波的叠加，但二者并不相同。干涉是指来自两个或多个相干光源（如双缝）的波的叠加；衍射则是由于惠更斯原理，单个光源或孔缝中每一点都成为新的子波源，从而导致波的扩展。

Double-slit interference: equally spaced bright fringes of similar intensity (for small angles).
Single-slit diffraction: a broad central maximum with rapidly decreasing intensity and unequally spaced fringes.
Combined effect: in a double-slit experiment, each slit produces diffraction, and the two sets of waves interfere, creating an interference pattern modulated by the diffraction envelope.

双缝干涉：明纹等间距且强度相近（小角度下）。
单缝衍射：中央亮纹宽，两侧亮纹强度迅速衰减，条纹间距不等。
综合效果：在双缝实验中，每个缝都产生衍射，而两列波发生干涉，形成由衍射包络调制的干涉图样。

7. Observing Diffraction in the Laboratory | 实验室中观察衍射

A common setup uses a laser (monochromatic, coherent) directed at a single adjustable slit, with a screen positioned a few metres away. The observed pattern shows a bright central band, with weaker side bands. Using a white light source instead produces overlapping coloured patterns because different wavelengths diffract by different amounts.

常见的实验装置使用激光（单色、相干）照射可调宽度的单缝，屏幕置于数米之外。观察到的图样显示一条明亮的中央带和较弱的侧带。若改用白光光源，则因不同波长的衍射程度不同，会产生彩色重叠图样。

To measure wavelength using diffraction, one can use a diffraction grating (many parallel slits) with a known slit spacing d. The grating equation is:

d sin θ = mλ, m = 0, 1, 2, …

要利用衍射测量波长，可以使用已知缝间距 d 的光栅（大量平行狭缝）。光栅方程为：

d sin θ = mλ，m = 0，1，2，…

The grating produces sharp, widely spaced principal maxima, making it easier to measure angles accurately than with a single slit.

光栅产生细锐、间距较大的主极大，比单缝更容易精确测量角度。

8. Common Question Types | 常见题型

Exam questions on light diffraction typically fall into four categories:

关于光衍射的考题通常分为四类：

Conceptual questions on conditions for diffraction, comparison with interference, and wave–particle duality.
Numerical problems using a sin θ = nλ to find slit width, wavelength, or angle.
Graphical or experimental questions involving intensity vs. angle graphs, or describing how a pattern changes when a variable is altered.
Resolution questions using the Rayleigh criterion for telescopes, cameras, or the human eye.

概念题：考查衍射条件、与干涉的比较以及波粒二象性。
计算题：利用 a sin θ = nλ 求缝宽、波长或角度。
图形/实验题：涉及光强–角度图，或描述改变某变量时图样的变化。
分辨类问题：用瑞利判据计算望远镜、相机或人眼的分辨极限。

9. Worked Example 1: Finding the Slit Width | 例题1：求缝宽

Problem: Monochromatic light of wavelength 600 nm is incident on a single slit. The first diffraction minimum is observed at an angle of 3.0° from the central axis. Determine the slit width.

题目：波长为 600 nm 的单色光照射单缝。第一级暗纹出现在距中心轴 3.0° 处。求缝宽。

Solution: For the first minimum, n = 1:

a sin θ = λ ⇒ a = λ / sin θ

Converting θ to degrees directly, sin 3.0° = 0.0523. Thus:

a = 600 × 10⁻⁹ / 0.0523 ≈ 1.15 × 10⁻⁵ m = 11.5 μm

解答：对于第一级暗纹，n = 1：

a sin θ = λ ⇒ a = λ / sin θ

sin 3.0° = 0.0523，因此：

a = 600 × 10⁻⁹ / 0.0523 ≈ 1.15 × 10⁻⁵ m = 11.5 μm

Check: The slit width (11.5 μm) is much larger than the wavelength (0.6 μm), which is consistent with the observable small-angle diffraction.

检验：缝宽（11.5 μm）远大于波长（0.6 μm），这与可观察的小角度衍射一致。

10. Worked Example 2: Width of the Central Maximum | 例题2：中央亮纹宽度

Problem: A slit of width 0.20 mm is illuminated by light of wavelength 500 nm. A screen is placed 2.0 m away. Calculate the width of the central maximum.

题目：缝宽为 0.20 mm，用波长 500 nm 的光照射。屏幕距缝 2.0 m。求中央亮纹的宽度。

Solution: The half-width is the distance from the centre to the first minimum:

y₁ = λL / a = (500 × 10⁻⁹ × 2.0) / (0.20 × 10⁻³) = 5.0 × 10⁻³ m

The full width of the central maximum is therefore:

Δy = 2y₁ = 1.0 × 10⁻² m = 1.0 cm

解答：半宽度即中央到第一级暗纹的距离：

y₁ = λL / a = (500 × 10⁻⁹ × 2.0) / (0.20 × 10⁻³) = 5.0 × 10⁻³ m

因此中央亮纹的全宽为：

Δy = 2y₁ = 1.0 × 10⁻² m = 1.0 cm

Note: If the slit width were halved to 0.10 mm, the central maximum would double to 2.0 cm. This inverse relationship is a frequent exam point.

注意：若缝宽减半至 0.10 mm，中央亮纹宽度将加倍至 2.0 cm。这种反比关系是常考要点。

11. Worked Example 3: Resolution of a Telescope | 例题3：望远镜的分辨率

Problem: A telescope has an objective lens of diameter 10 cm. It is used to observe light of wavelength 550 nm. What is the minimum angular separation that can be resolved?

题目：望远镜物镜直径为 10 cm，用于观察波长为 550 nm 的光。它能分辨的最小角间距是多少？

Solution: Using the Rayleigh criterion:

θ_min = 1.22 λ / D = 1.22 × (550 × 10⁻⁹) / (0.10) = 6.71 × 10⁻⁶ rad

Converting to arcseconds (1 rad = 206265 arcsec):

θ_min ≈ 6.71 × 10⁻⁶ × 206265 ≈ 1.38 arcsec

解答：利用瑞利判据：

θ_min = 1.22 λ / D = 1.22 × (550 × 10⁻⁹) / (0.10) = 6.71 × 10⁻⁶ rad

换算为角秒（1 rad = 206265 角秒）：

θ_min ≈ 6.71 × 10⁻⁶ × 206265 ≈ 1.38 角秒

Interpretation: Two stars closer than roughly 1.4 arcseconds would appear as a single blurred point through this telescope. A larger diameter lens improves resolution because θ_min decreases.

物理解释：若两颗星的角间距小于约 1.4 角秒，通过该望远镜观察时它们将看起来是一个模糊的光点。增大透镜直径可提高分辨率，因为 θ_min 会减小。

12. Exam Tips and Common Pitfalls | 考试技巧与常见误区

Students often lose marks on diffraction questions due to small but avoidable errors. Pay attention to the following:

学生在衍射题上失分往往是因为一些细小但可以避免的错误。请特别注意以下几点：

Use radians for small angles – The approximation sin θ ≈ θ only holds when θ is in radians, not degrees.
Identify n correctly – For single-slit minima, n starts at ±1; for grating maxima, m starts at 0.
Distinguish between y and Δy – A common mistake is giving the half-width when the question asks for the full width of the central maximum.
Unit conversions – Always convert nm to m and mm to m before substitution; forgetting to do so leads to errors of 10⁶ or more.
Do not confuse single-slit diffraction with double-slit interference – The formulas and fringe patterns are different; read the question carefully.

小角度必须用弧度：sin θ ≈ θ 仅在 θ 以弧度为单位时成立，而不是度数。
正确确定 n：单缝暗纹 n 从 ±1 开始；光栅主极大 m 从 0 开始。
区分 y 与 Δy：常见错误是将半宽当作题目所要求的中央亮纹全宽。
单位换算：代入前务必把 nm 换成 m、mm 换成 m；忘记换算会导致 10⁶ 倍甚至更大的误差。
不要混淆单缝衍射与双缝干涉：两者的公式和条纹图样不同；务必仔细审题。

Finally, remember that diffraction is a wave phenomenon. The fact that light can bend around obstacles and produce interference patterns was historically crucial in establishing the wave theory of light, and it remains a favourite topic for examiners seeking to test both conceptual understanding and quantitative skill.

最后请记住，衍射是波动现象。光能绕过障碍物并产生干涉图样，这在历史上对确立光的波动理论至关重要，如今也一直是命题者考查概念理解与定量计算能力的热门主题。

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Light Diffraction Phenomena and Common Question Types | 光的衍射现象与常见题型