📚 Diffraction Grating: Principle and Formulas | 衍射光栅的工作原理与公式
A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams travelling in different directions. It is one of the most important tools in spectroscopy because it allows precise measurement of wavelength and spectral lines. In this article, we explain the working principle, derive the grating equation, and discuss key calculations required by the CIE A-Level syllabus.
衍射光栅是一种具有周期性结构的光学元件，它能把光分割并衍射成沿不同方向传播的多束光。由于它能精确测量波长和光谱线，因此是光谱学中最重要的工具之一。在本文中，我们将解释其工作原理、推导光栅方程，并讨论 CIE A-Level 教学大纲要求的关键计算。
1. Grating Structure and Grating Constant | 光栅的结构与光栅常数
A transmission grating consists of a very large number of equally spaced parallel slits, typically thousands of lines per millimetre. The distance between adjacent slits is called the grating spacing or grating constant, usually denoted by d. If a grating has N lines per metre, then d = 1/N. It is essential to convert the quoted “lines per mm” into metres before making calculations.
透射式光栅由大量等间距的平行狭缝组成，通常每毫米有数千条刻线。相邻狭缝之间的距离称为光栅间距或光栅常数，常用 d 表示。若光栅每米有 N 条刻线，则 d = 1/N。在计算之前，务必把“每毫米刻线数”换算成以米为单位的 d。
For example, a grating with 300 lines per millimetre has grating spacing d = 1/300 mm = 3.33 × 10⁻⁶ m = 3.33 μm. The smaller the grating spacing, the more widely the spectral lines are spread. A grating with 600 lines per millimetre gives d = 1.67 × 10⁻⁶ m, producing a larger diffraction angle for the same wavelength.
例如，每毫米 300 条刻线的光栅，其光栅间距为 d = 1/300 mm = 3.33 × 10⁻⁶ m = 3.33 μm。光栅常数越小，光谱线展开得越宽。每毫米 600 条刻线的光栅给出 d = 1.67 × 10⁻⁶ m，对同一波长产生的衍射角更大。
2. Working Principle: Multiple-Slit Interference | 工作原理：多缝干涉
When monochromatic light is incident normally on a grating, each slit acts as a secondary source of waves according to Huygens’ principle. The waves from all slits overlap and superpose. In most directions, the waves cancel by destructive interference, producing darkness. However, in certain directions the waves from all slits arrive exactly in phase and undergo constructive interference, producing a bright principal maximum.
当单色光垂直入射到光栅上时，根据惠更斯原理，每一个狭缝都充当一个次级波源。来自所有狭缝的波相互重叠与叠加。在大多数方向上，波因相消干涉而互相抵消，形成暗区。但在某些特定方向上，来自所有狭缝的波恰好同相到达，发生相长干涉，从而产生明亮的主极大。
The key idea is that the grating has many slits, not just two. With more slits, the principal maxima become sharper, brighter, and more widely separated by broad dark regions. This is why a grating can measure wavelength far more accurately than a double slit.
关键之处在于光栅包含大量狭缝，而不仅仅是两条。狭缝越多，主极大就越尖锐、越明亮，而且被更宽的暗区分隔开。这就是光栅测量波长远比双缝精确的原因。
3. Derivation of the Grating Equation | 光栅方程的推导
Consider two adjacent slits separated by d. A ray from the upper slit and a ray from the lower slit travel at an angle θ to the normal. The extra distance travelled by the lower ray is d sin θ. This is the path difference between waves from adjacent slits.
取两个相距 d 的相邻狭缝。来自上方狭缝和下方狭缝的光线沿与法线成 θ 角的方向传播。下方光线多走的路程为 d sin θ，这就是相邻狭缝发出波的光程差。
Path difference = d sin θ
光程差 = d sin θ
Constructive interference happens when this path difference equals an integer multiple of the wavelength. Hence the grating equation is:
当光程差等于波长的整数倍时发生相长干涉。因此光栅方程为：
d sin θ = n λ, n = 0, ±1, ±2, ±3, …
d sin θ = n λ，n = 0, ±1, ±2, ±3, …
Here θ is the diffraction angle measured from the normal to the grating, n is the order of the principal maximum, and λ is the wavelength of the light. For n = 0, θ = 0, so the zeroth-order maximum is always straight ahead; it contains all wavelengths superimposed and shows no dispersion.
其中 θ 是相对于光栅法线测量的衍射角，n 是主极大的级次，λ 是光的波长。对于 n = 0，θ = 0，因此零级最大总在正前方；它包含所有波长叠加在一起，不显示色散。
4. Maximum Order of Diffraction | 最大衍射级次
Since sin θ can never exceed 1, the grating equation imposes a limit on the observable orders. For a given d and λ, the maximum order nmax is the largest integer that satisfies nmax < d/λ.
由于 sin θ 永远不能大于 1，光栅方程对可观察的级次施加了限制。对于给定的 d 和 λ，最大级次 nₘₐₓ 是满足 nₘₐₓ < d/λ 的最大整数。
For example, if d = 2.0 × 10⁻⁶ m and λ = 450 nm = 4.5 × 10⁻⁷ m, then d/λ = 2.0 × 10⁻⁶ ÷ 4.5 × 10⁻⁷ = 4.44. The highest order visible is therefore n = 4. An order n = 5 would require sin θ = 2.22, which is impossible. If d/λ is less than 1, only the zeroth order exists and no spectral separation occurs.
例如，d = 2.0 × 10⁻⁶ m、λ = 450 nm = 4.5 × 10⁻⁷ m 时，d/λ = 2.0 × 10⁻⁶ ÷ 4.5 × 10⁻⁷ = 4.44。因此可见的最高级次是 n = 4。若取 n = 5，则要求 sin θ = 2.22，这是不可能的。若 d/λ 小于 1，则只存在零级，不发生光谱分离。
This limit is frequently tested. You should always check whether the requested order is actually physically observable before computing the angle.
这一限制是常见考点。在计算角度之前，应始终检查所要求级次在物理上是否真的可观测。
5. Angle of Diffraction and Order Relationship | 衍射角与级次关系
Rearranging the grating equation gives sin θ = nλ/d. For a fixed wavelength, higher orders correspond to larger angles. The first-order angle satisfies sin θ₁ = λ/d; the second-order angle satisfies sin θ₂ = 2λ/d = 2 sin θ₁. Notice that the angles are not evenly spaced because sine is not proportional to angle.
将光栅方程变形可得 sin θ = nλ/d。在波长一定时，级次越高，角度越大。一级角满足 sin θ₁ = λ/d；二级角满足 sin θ₂ = 2λ/d = 2 sin θ₁。注意各角度并不是均匀间隔的，因为正弦并不与角度成正比。
When a beam is normally incident and you measure the positions of maxima on a screen at distance L from the grating, you can use tan θ = y/L, where y is the lateral distance from the central maximum. Combining sin θ = nλ/d with tan θ = y/L allows d or λ to be found from measured y values. A graph of sin θ against n for a fixed wavelength should be a straight line through the origin with gradient λ/d.
当光束垂直入射，并测量距光栅为 L 的屏幕上极大位置时，可用 tan θ = y/L，其中 y 是距中心极大的横向距离。将 sin θ = nλ/d 与 tan θ = y/L 结合，可由测得的 y 值求出 d 或 λ。在波长一定时，作出 sin θ 随 n 变化的图线，应为过原点且斜率为 λ/d 的直线。
6. White Light and Dispersion | 白光与色散
When white light passes through a diffraction grating, each order becomes a continuous spectrum. Because θ depends on λ, violet light is diffracted least and red light is diffracted most in each order. The first-order spectrum ranges from violet at small angle to red at larger angle; higher orders repeat this pattern with wider angular spread.
当白光通过衍射光栅时，每一级都成为连续光谱。由于 θ 取决于 λ，每一级中紫光衍射最小、红光衍射最大。一级光谱从角度较小的紫光延伸到角度较大的红光；更高阶次重复这一模式，且角度展开更宽。
Overlap can occur when the red end of one order coincides with the violet end of the next order. In general, the condition for no overlap between order n and order n + 1 is nλred < (n + 1)λviolet. If this is not satisfied, the spectrum becomes ambiguous. This idea is a common examination question involving numerical comparisons.
当一个级次的红端与下一级次的紫端重合时，会发生重叠。一般来说，第 n 级与第 n + 1 级不发生重叠的条件是 nλ红 < (n + 1)λ紫。若不满足该条件，光谱将变得难以分辨。这是考试中常见的一类涉及数值比较的问题。
7. Resolving Power | 分辨本领
The resolving power of a grating describes its ability to distinguish two wavelengths that are very close together, such as the sodium doublet at 589.0 nm and 589.6 nm. For a grating with a total of N illuminated slits, working in order n, the resolving power is:
光栅的分辨本领描述的是它分辨两个波长非常接近的光线（如钠双线 589.0 nm 和 589.6 nm）的能力。对于总共被照亮的狭缝数为 N、工作级次为 n 的光栅，分辨本领为：
R = N n = λ/Δλ
R = N n = λ/Δλ
where Δλ is the smallest wavelength difference that can be resolved near wavelength λ. To resolve the two sodium lines with λ = 589 nm and Δλ = 0.6 nm, we need R = 589/0.6 ≈ 982. If the grating is used in the second order, this requires N ≥ 491 illuminated slits.
其中 Δλ 是在波长 λ 附近可分辨的最小波长差。要分辨 λ = 589 nm、Δλ = 0.6 nm 的钠双线，需要 R = 589/0.6 ≈ 982。若用二级光谱观察，则需要被照亮的狭缝数 N ≥ 491。
Notice that R increases with the order n and with the total number of slits illuminated by the beam. Widening the beam so that it covers more lines, or using a grating with more lines per millimetre, improves resolution. However, increasing N also increases the grating spacing d if the ruled width is fixed, which may reduce the available order.
注意 R 随级次 n 和光束照亮的狭缝总数 N 增大而增大。加宽光束使其覆盖更多刻线，或使用每毫米刻线更多的光栅，能提高分辨率。但如果刻划宽度固定，增大 N 也会增大总刻线数对 d 的影响，这可能降低可获得的最大级次。
8. Comparison with Double-Slit Interference | 与双缝干涉的比较
Both a double slit and a diffraction grating produce interference maxima from the same condition d sin θ = nλ, but their fringe patterns are very different. In Young’s double-slit experiment, the intensity gradually falls off and the maxima are broad because only two waves interfere.
双缝与衍射光栅都通过 d sin θ = nλ 这一条件产生干涉极大，但它们的条纹图样差别很大。在杨氏双缝实验中，由于只有两列波干涉，极大较宽，强度逐渐变化。
With a grating containing thousands of slits, the principal maxima are extremely sharp and the dark regions between them are very wide. For equal d and λ, the angular positions of the maxima are the same, but the grating pattern is much more precise. This makes the grating far superior for measuring wavelengths and analysing spectra.
而含有数千条狭缝的光栅，其主极大极为尖锐，相邻主极大之间的暗区非常宽。在 d 和 λ 相同时，双缝与光栅的极大角度位置相同，但光栅的图样精确得多。因此光栅在波长测量与光谱分析方面远优于双缝。
The table below summarises the key differences:
下表总结了主要区别：
| Feature | 特征 | Double slit | 双缝 | Diffraction grating | 衍射光栅 |
| Number of slits | 狭缝数 | 2 | Thousands | 数千条 |
| Maxima sharpness | 极大锐度 | Broad | 宽 | Very sharp | 极尖锐 |
| Brightness | 亮度 | Low | 较低 | High | 较高 |
| Use | 用途 | Measuring wavelength | 测波长 | Spectroscopy | 光谱分析 |
9. Experimental Measurement and Precision | 实验测量与精确度
In the standard CIE practical, a monochromatic light source or laser illuminates a grating mounted perpendicular to the beam. The diffracted maxima are observed on a screen, and the angle θ is found using tan θ = y/L. Because the maxima are so sharp, y can be measured with high precision, and the uncertainty in λ is dominated by the uncertainty in L and the perpendicular alignment of the grating.
在标准 CIE 实验中，单色光源或激光垂直照射光栅。衍射极大在屏上观察，并通过 tan θ = y/L 求出角度 θ。由于主极大非常锐利，y 的测量精度很高，λ 的不确定度主要来自 L 的测量不确定度以及光栅是否严格垂直。
To reduce systematic error, measure the positions of the maxima on both sides of the central maximum and take the average. Also use the largest possible value of L so that the fractional uncertainty in L is small. Ensure the grating surface is perpendicular to the beam; if it is tilted, the effective d along the beam direction changes and the simple grating equation with the nominal d no longer applies.
为减小系统误差，应测量中央极大两侧的级次位置并取平均值。同时，应尽量增大 L，使 L 的相对不确定度变小。务必确保光栅表面垂直于光束；若光栅倾斜，光束方向上的等效 d 会
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