Diffraction of Light | 光的衍射

📚 Diffraction of Light | 光的衍射

Diffraction is one of the most fascinating wave phenomena, revealing how light bends around obstacles and spreads through narrow openings. In IB and CCEA Physics, mastering diffraction means understanding not only the patterns we observe but also the underlying principles that connect wave theory to real-world optical instruments. This article provides a clear, exam-focused guide to the key concepts, equations, and common pitfalls in the topic of light diffraction.

衍射是最迷人的波动现象之一，它揭示了光如何绕过障碍物并在狭窄缝隙中扩展。在 IB 和 CCEA 物理中，掌握衍射不仅意味着理解我们观察到的图样，还意味着理解将波动理论与实际光学仪器挂钩的基本原理。本文围绕考试重点，清晰梳理光衍射的核心概念、关键方程和常见易错点。

1. What is Diffraction? | 什么是衍射？

Diffraction is the spreading of waves when they pass through a narrow aperture or around an obstacle. This effect is most pronounced when the size of the aperture or obstacle is comparable to the wavelength of the wave. For light, with wavelengths in the range of 400–700 nm, diffraction becomes significant only with very small openings, such as a single slit a fraction of a millimetre wide.

衍射是波在穿过窄缝或绕过障碍物时扩展的现象。当缝隙或障碍物的尺寸与波长相近时，衍射效应最为显著。对于波长在 400–700 nm 范围的光，只有在非常小的开口（例如宽度不到一毫米的单缝）下，衍射才变得明显。

2. Huygens–Fresnel Principle | 惠更斯–菲涅耳原理

Huygens’ principle states that every point on a wavefront acts as a source of secondary spherical wavelets. Fresnel later added that the amplitude of the wave at any point is the superposition of all these wavelets. This principle successfully explains why light bends into the geometric shadow region and produces a diffraction pattern on a screen.

惠更斯原理指出，波前上的每一点都可以看作次球面子波的波源。菲涅耳随后补充，空间任意一点的振幅是所有子波叠加的结果。这一原理成功解释了光为什么会弯曲进入几何阴影区并在屏上产生衍射图样。

3. Single-Slit Diffraction Setup | 单缝衍射实验装置

A typical setup uses a monochromatic light source, such as a laser, directed at a narrow rectangular slit of width a. The light passing through the slit falls onto a screen placed at a large distance D compared with the slit width. The resulting pattern consists of a bright central maximum flanked by alternating dark and bright fringes of decreasing intensity.

典型装置使用单色光源（如激光）照射一个宽度为 a 的窄矩形狭缝。通过狭缝的光投射到距离狭缝很远（与缝宽相比）的屏幕上。产生的图样由一个明亮的中央主极大和两侧交替出现的暗条纹和亮条纹组成，强度逐渐减弱。

4. Single-Slit Intensity Pattern | 单缝衍射强度图样

The central maximum is twice as wide as the secondary maxima and much brighter. Minima occur at angles θ satisfying a sin θ = nλ, where n = 1, 2, 3, … and λ is the wavelength. The intensity I at an angle θ relative to the central maximum is given by I = I₀ (sin β / β)², where β = (π a sin θ)/λ. Secondary maxima lie approximately midway between the minima.

中央主极大的宽度是次极大两倍，且亮得多。极小值出现在满足 a sin θ = nλ 的角度 θ 上，其中 n = 1, 2, 3, … ，λ 为波长。相对中央主极大的强度 I 由 I = I₀ (sin β / β)² 给出，其中 β = (π a sin θ)/λ。次极大大致位于相邻暗纹的中间。

a sin θ = nλ (n = 1, 2, 3, …)

5. Effect of Slit Width and Wavelength | 缝宽和波长的影响

The angular width of the central maximum is 2λ / a in radians for small angles. Therefore, decreasing the slit width a increases the spread of the diffraction pattern, while decreasing the wavelength makes the pattern narrower. Using white light produces a central white fringe and coloured side fringes because different wavelengths diffract by different amounts.

在小角度下，中央主极大的角宽度为 2λ / a 弧度。因此，减小缝宽 a 会增大衍射图样的扩展程度，而减小波长则使图样变窄。使用白光时，中央条纹为白色，两侧出现彩色条纹，因为不同波长的光衍射角度不同。

6. Diffraction Grating | 衍射光栅

A diffraction grating consists of a large number of equally spaced parallel slits. The distance between adjacent slits, d, is called the grating spacing. When monochromatic light is incident on a grating, constructive interference occurs at angles θ given by the grating equation. Gratings produce much sharper and brighter maxima than a double slit, making them ideal for measuring wavelengths precisely.

衍射光栅由大量等间距的平行刻线组成。相邻刻线间的距离 d 称为光栅常数。当单色光照射光栅时，在满足光栅方程的角度 θ 处出现相长干涉。光栅产生的亮纹比双缝锐利明亮得多，因此非常适合精确测量波长。

7. The Grating Equation | 光栅方程

The condition for principal maxima in a transmission grating is d sin θ = nλ, where n is the order of diffraction (0, 1, 2, …). The zeroth order (n = 0) is the straight-through beam. The number of slits per unit length N is related by d = 1/N. For a given grating, higher orders appear at larger angles, and a maximum number of orders exists when sin θ ≤ 1.

透射光栅产生主极大的条件是 d sin θ = nλ，其中 n 为衍射级次（0, 1, 2, …）。零级（n = 0）为直射光束。单位长度的刻线数 N 与光栅常数的关系为 d = 1/N。对给定的光栅，更高级次的亮纹出现在更大的角度，由于 sin θ ≤ 1，可观察到的级次存在最大值。

d sin θ = nλ (n = 0, 1, 2, …)

8. Using a Grating to Measure Wavelength | 利用光栅测量波长

By measuring the angles of diffraction orders and knowing the grating spacing, the wavelength of light can be calculated using λ = d sin θ / n. In the lab, a spectrometer or a simple setup with a metre ruler can be used. Care must be taken to measure the angle of each order on both sides of the central maximum and average them to reduce errors.

通过测量各级衍射角度并已知光栅常数，可利用 λ = d sin θ / n 计算光波长。在实验室中，可使用分光计或简单的米尺装置进行测量。应注意测量中央主极大两侧各级的角度，并取平均值以减小误差。

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

Although both phenomena arise from superposition, diffraction refers specifically to the bending and spreading of waves due to an aperture or obstacle, while interference describes the combination of two or more separate coherent sources. In a double-slit experiment, the overall pattern is the product of single-slit diffraction and two-slit interference, giving rise to a modulated fringe system.

虽然两种现象都源于叠加，但衍射特指波因开口或障碍物而产生的弯曲和扩展，而干涉描述的是两个或多个独立相干源波动的叠加。在双缝实验中，总图样是单缝衍射与双缝干涉的乘积，从而产生一个被调制的条纹系统。

10. Rayleigh Criterion and Resolution | 瑞利判据与分辨率

The Rayleigh criterion states that two point sources are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other. The angular resolution limit is θ_min ≈ 1.22λ / D for a circular aperture of diameter D. This concept is crucial in understanding the resolving power of telescopes, microscopes, and the human eye.

瑞利判据指出，当一个衍射图样的中央主极大恰好落在另一个图样的第一极小值时，两个点源刚好能够分辨。对于直径为 D 的圆孔，角分辨率极限为 θ_min ≈ 1.22λ / D。这一概念对理解望远镜、显微镜和人眼的分辨本领至关重要。

11. Real-World Applications of Diffraction | 衍射的实际应用

Diffraction is not only a laboratory curiosity. It is used in X-ray crystallography to determine atomic structures, in the design of holograms, and in the analysis of light from stars. Even the limits of optical storage media such as Blu-ray discs are set by the diffraction limit of the laser used to read the data.

衍射不仅是实验室中的奇观，它在 X 射线晶体学中用于确定原子结构，在全息图设计以及恒星光谱分析中都有应用。即便是蓝光光盘等光学存储介质的容量极限，也是由读取数据所用激光的衍射极限决定的。

12. Exam Tips and Common Mistakes | 备考提示与常见错误

Always distinguish between single-slit minima condition (a sin θ = nλ) and grating maxima condition (d sin θ = nλ). Confusing the slit width a with the grating spacing d costs many marks. | 始终分清单缝极小条件（a sin θ = nλ）和光栅极大条件（d sin θ = nλ）。将缝宽 a 与光栅常数 d 混淆会丢很多分。
When drawing diffraction patterns, the central maximum must be drawn with twice the width of secondary maxima and with much greater amplitude. | 画衍射图样时，中央主极大的宽度必须画成次极大的两倍，且振幅要大得多。
For gratings, students often forget that the number of orders visible is limited by sin θ ≤ 1. Solve n_max = d/λ and take the integer part. | 对于光栅，学生常忘记可见级次受限于 sin θ ≤ 1。计算 n_max = d/λ 并向下取整。
In interference and diffraction superposed problems, recognise that the double-slit fringe spacing Δy = λD/s is modulated by the single-slit envelope. | 在干涉与衍射叠加的问题中，要认识到双缝条纹间距 Δy = λD/s 被单缝衍射包络所调制。

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Diffraction of Light | 光的衍射