📚 IB OCR Physics: Light Diffraction Key Points | IB OCR物理：光的衍射考点精讲
Diffraction is a fundamental wave phenomenon that occurs when light encounters an obstacle or a slit. Understanding diffraction is essential for IB and OCR physics students, as it not only explains the behavior of light beyond geometrical optics but also underpins many modern technologies such as spectroscopy and high-resolution imaging.
衍射是光遇到障碍物或狭缝时发生的一种基本波动现象。理解衍射对于 IB 和 OCR 物理学生至关重要，因为它不仅解释了超越几何光学的光的行为，而且是光谱学和高分辨率成像等许多现代技术的基础。
1. What is Diffraction? | 什么是衍射？
Diffraction refers to the spreading of waves as they pass through an aperture or around an obstacle. The amount of bending depends on the ratio of the wavelength to the size of the opening. For light, significant diffraction occurs when the slit width is comparable to the wavelength, typically a few micrometres. If the aperture is much larger than the wavelength, the wave travels in straight lines and geometrical optics applies.
衍射是指波通过孔径或绕过障碍物时发生的扩展。弯曲的程度取决于波长与开口大小的比值。对于光，当缝宽与波长（通常几微米）相当时，会发生显著的衍射。如果孔径远大于波长，波将沿直线传播，几何光学适用。
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 wavefront is the envelope of these wavelets. When the wavefront encounters an aperture, the wavelets along the edges propagate into the geometrical shadow, explaining diffraction. The superposition of these secondary wavelets gives rise to the interference pattern observed on a screen.
惠更斯原理指出波前上的每一点都可以视为次波的点源。新的波前是这些次波的包络面。当波前遇到孔径时，边缘处的次波会传播到几何阴影区，从而解释了衍射现象。这些次波的叠加在屏幕上形成了观察到的干涉图样。
3. Single-Slit Diffraction: Intensity Pattern | 单缝衍射的强度分布
When monochromatic light passes through a narrow single slit of width a, a diffraction pattern appears on a distant screen. It consists of a broad central bright fringe (the central maximum) flanked by alternating dark and bright fringes of decreasing intensity. The intensity I as a function of angle θ can be described by the expression I = I₀ (sinβ / β)², where β = (π a sinθ) / λ and I₀ is the intensity at the centre. Most exam questions focus on the positions of dark fringes and the angular width of the central maximum.
当单色光通过宽度为 a 的窄单缝时，远处的屏幕上会出现衍射图样。它包括一个宽阔的中央亮纹（中央极大），两侧是强度逐渐减弱的交替明暗条纹。强度 I 随角度 θ 的变化可以表示为 I = I₀ (sinβ / β)²，其中 β = (π a sinθ) / λ，I₀ 是中心处的强度。大多数考题关注暗纹的位置和中央明纹的角宽度。
The positions of dark fringes (minima) are given by the single-slit equation:
a sinθ = mλ , m = ±1, ±2, ±3, …
Note that m = 0 corresponds to the central maximum, not a dark fringe. For small angles, the angular width of the central maximum is approximately Δθ = 2λ / a, which shows that a narrower slit produces a wider central maximum.
暗纹（极小）的位置由单缝方程给出：a sinθ = mλ，m = ±1, ±2, ±3, …。注意 m = 0 对应中央极大而不是暗纹。对于小角度，中央明纹的角宽度近似为 Δθ = 2λ / a，这表明缝越窄，中央明纹越宽。
4. Factors Affecting the Diffraction Pattern | 影响衍射图样的因素
The single-slit diffraction pattern is highly dependent on the slit width a and the wavelength λ. When a decreases, the angle θ for the first minimum (m=1) increases, so the pattern spreads out. Similarly, a larger wavelength λ broadens the pattern. This is why red light (longer λ) diffracts more than blue light (shorter λ) through the same slit. In exam scenarios, you may be asked to sketch how the pattern changes when the slit width is halved or when white light is used.
单缝衍射图样很大程度上取决于缝宽 a 和波长 λ。当 a 减小时，第一极小（m=1）对应的角度 θ 增大，因此图样发散。同样，较大的波长 λ 也会使图样变宽。这就是为什么红光（λ 较长）通过相同狭缝时比蓝光（λ 较短）衍射更明显。在考试中，你可能会被要求画出当缝宽减半或使用白光时图样的变化。
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 grooves). The distance between adjacent slits is d, called the grating spacing. When monochromatic light is incident on the grating, constructive interference produces very sharp and bright peaks called principal maxima at angles given by the grating equation:
d sinθ = nλ , n = 0, ±1, ±2, …
Here n is the order of diffraction. The zeroth order (n=0) is the straight-through beam, and higher orders appear symmetrically on either side. The grating produces much sharper lines compared to a double slit because multiple slits reinforce the interfering beams.
衍射光栅由大量等间距的平行狭缝（或刻线）组成。相邻狭缝的间距为 d，称为光栅常数。当单色光照射光栅时，相长干涉会在满足光栅方程 d sinθ = nλ (n = 0, ±1, ±2, …) 的角度上产生非常锐利且明亮的峰，称为主极大。n 是衍射级次。零级 (n=0) 是直射光束，高级次对称出现在两侧。与双缝相比，光栅产生的谱线要锐利得多，因为多重狭缝增强了干涉光束。
6. Angular Dispersion and Resolving Power of a Grating | 光栅的角色散与分辨本领
The angular dispersion measures how much the angular position of a maximum changes with wavelength. For a grating, it is derived from differentiating the grating equation: dθ/dλ = n / (d cosθ). This means better dispersion is achieved with a higher order n and a smaller grating spacing d. The resolving power R is the ability to separate two closely spaced wavelengths, λ and λ + Δλ, defined as R = λ / Δλ. For a grating, R = nN, where N is the total number of illuminated lines (or slits). A higher number of lines and a higher diffraction order increase the resolving power.
角色散衡量最大值的角位置随波长的变化率。对光栅而言，由光栅方程微分得到 dθ/dλ = n / (d cosθ)。这意味着更高的级次 n 和更小的光栅常数 d 能获得更好的色散。分辨本领 R 是指区分两个非常接近的波长 λ 和 λ + Δλ 的能力，定义为 R = λ / Δλ。对于光栅，R = nN，其中 N 是被照射的总线条数（或狭缝数）。线条数越多，衍射级次越高，分辨本领越强。
7. Double-Slit Interference with Diffraction Effects | 双缝干涉与衍射效应
In an ideal Young’s double-slit experiment, the slits are treated as infinitely narrow, producing equally bright interference fringes. In reality, each slit has a finite width a, and its single-slit diffraction pattern modulates the interference fringes. The resulting intensity pattern is the product of the double-slit interference term and the single-slit diffraction envelope. The interference maxima still occur at d sinθ = mλ (m = 0, ±1, ±2, …) but their brightness is scaled by the single-slit intensity function. When a maximum of the interference coincides with a minimum of the diffraction envelope, that fringe vanishes — this is known as a missing order.
在理想的双缝干涉实验中，狭缝被视为无限窄，产生亮度相等的干涉条纹。实际上，每条缝都有有限的宽度 a，其单缝衍射图样会调制干涉条纹。最终的强度图样是双缝干涉项与单缝衍射包络的乘积。干涉极大仍然出现在 d sinθ = mλ (m = 0, ±1, ±2, …) 处，但其亮度受到单缝强度函数的缩放。当干涉极大与衍射包络的极小重合时，该条纹就会消失——这就是缺级现象。
8. Missing Orders Explained | 缺级解释
A missing order occurs when the path difference condition for an interference maximum, d sinθ = mλ, coincides with the condition for a diffraction minimum, a sinθ = nλ (n = ±1, ±2, …). Dividing the two equations gives the condition for a missing order: d / a = m / n, where m and n are integers. For example, if the slit separation d is twice the slit width a (d = 2a), then interference orders m = 2, 4, 6, … correspond to diffraction minima n = 1, 2, 3, …, so every even-order bright fringe disappears.
当干涉极大的路径差条件 d sinθ = mλ 与衍射极小的条件 a sinθ = nλ (n = ±1, ±2, …) 重合时，就出现缺级。将两式相除，可得缺级条件：d / a = m / n，其中 m 和 n 都是整数。例如，如果缝间距 d 是缝宽 a 的两倍（d = 2a），那么干涉级次 m = 2, 4, 6, … 对应衍射极小 n = 1, 2, 3, …，因此每个偶数级亮纹都会消失。
9. Diffraction by a Circular Aperture: the Airy Disk | 圆孔衍射与艾里斑
When light passes through a circular aperture of diameter D, the diffraction pattern is characterized by a central bright region called the Airy disk, surrounded by concentric dark and bright rings. The angular position of the first dark ring is given by sinθ ≈ 1.22λ / D. For small angles, θ ≈ 1.22λ / D. This result is extremely important in optical instruments because the finite size of the Airy disk limits the resolution of the system.
当光通过直径为 D 的圆孔时，衍射图样由一个称为艾里斑的中央亮区以及围绕它的同心明暗环组成。第一暗环的角位置由 sinθ ≈ 1.22λ / D 给出。对于小角度，θ ≈ 1.22λ / D。这一结果在光学仪器中极其重要，因为艾里斑的有限大小限制了系统的分辨率。
10. Rayleigh Criterion and Optical Resolution | 瑞利判据与光学分辨率
The Rayleigh criterion provides a standard for when two adjacent point sources can be resolved. For a circular aperture, two sources are said to be just resolvable when the central maximum of one diffraction pattern falls exactly on the first minimum of the other. The minimum resolvable angular separation is thus θR = 1.22λ / D. This principle applies to telescopes, microscopes, and even the human eye. To improve resolution, you can increase the aperture diameter D or use a shorter wavelength λ (e.g., electron microscopes).
瑞利判据提供了一个判断两个相邻点光源能否被分辨的标准。对于圆孔，当一个衍射图样的中央极大刚好落在另一个衍射图样的第一极小时，就说这两个点光源刚好能被分辨。因此最小可分辨角为 θR = 1.22λ / D。这一原理适用于望远镜、显微镜，甚至人眼。要提高分辨率，可以增大孔径直径 D 或使用更短的波长 λ（例如电子显微镜）。
11. Common Exam Pitfalls and Tips | 常见考试陷阱与技巧
Students often confuse the single-slit equation (a sinθ = mλ) with the grating equation (d sinθ = nλ). Remember that in the single-slit pattern, the central maximum is at m = 0 and is the brightest, whereas the grating zeroth order is simply the undiffracted beam. When sketching double-slit patterns with finite slit widths, always draw the diffraction envelope limiting the fringe heights. Also, check whether the question asks for angular separation or linear separation on a screen, and be ready to use the small-angle approximation sinθ ≈ tanθ ≈ θ (in radians) if the distance to the screen is large.
学生经常混淆单缝方程 (a sinθ = mλ) 与光栅方程 (d sinθ = nλ)。请记住，在单缝图样中，中央极大位于 m = 0 且最亮，而光栅零级只是未被衍射的光束。在绘制有限缝宽的双缝图样时，一定要画出限制条纹高度的衍射包络。此外，要检查题目所问的是角间距还是屏幕上的线性间距，如果距离屏幕很远，准备好使用小角度近似 sinθ ≈ tanθ ≈ θ（弧度制）。
12. Summary and Key Formulae | 总结与关键公式
| Concept | Formula | Notes |
|---|---|---|
| Single-slit minima | a sinθ = mλ | m = ±1, ±2, …; central max at m=0 |
| Grating maxima | d sinθ = nλ | n = 0, ±1, ±2, … |
| Double-slit (ideal) | d sinθ = mλ | m = 0, ±1, ±2, … |
| Circular aperture (1st min) | sinθ ≈ 1.22λ / D | D = diameter |
| Rayleigh criterion | θR = 1.22λ / D | Just
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