📚 Diffraction gratings | 衍射光栅
Diffraction gratings are fundamental tools in optical physics and spectroscopy. They consist of a large number of equally spaced parallel slits or reflecting grooves that produce sharp interference maxima when illuminated with coherent or quasi-monochromatic light. By analyzing the angles at which constructive interference occurs, scientists can determine the wavelength of light with high precision. This article covers the essential concepts of diffraction gratings for CIE A-Level Physics, including the grating equation, angular dispersion, resolving power, and common experimental methods.
衍射光栅是光学物理和光谱学中的基本工具。它们由大量等间距的平行狭缝或反射刻槽构成，当用相干光或准单色光照射时，会产生尖锐的干涉极大。通过分析出现相长干涉的角度，科学家可以高精度地测定光的波长。本文涵盖 CIE A-Level 物理中衍射光栅的核心概念，包括光栅方程、角色散、分辨本领以及常见实验方法。
1. What is a Diffraction Grating? | 什么是衍射光栅？
A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams travelling in different directions. The key feature is a series of many equally spaced, identical diffracting elements – either transparent slits on an opaque background (transmission grating) or reflective grooves on a mirrored surface (reflection grating).
衍射光栅是一种具有周期性结构的光学元件，它能将光分解并衍射成多个沿不同方向传播的光束。其关键特征是一系列大量等间距、完全相同的衍射单元——可以是不透明背景上的透明狭缝（透射光栅），也可以是镜面上的反射刻槽（反射光栅）。
The distance between the centres of adjacent slits or grooves is called the grating spacing, symbolised by d. If a grating has N lines per metre, then d = 1/N. Commercial gratings often have 300, 600, or 1200 lines per millimetre, giving very small values of d.
相邻狭缝或刻槽中心之间的距离称为光栅常数，用符号 d 表示。如果光栅每米有 N 条刻线，则 d = 1/N。商品光栅通常具有每毫米 300、600 或 1200 条刻线，因此 d 的值非常小。
Unlike a double slit, a diffraction grating produces much sharper and brighter maxima because the constructive interference from many slits concentrates the light into very narrow angular regions. This sharpness makes gratings ideal for spectroscopy.
与双缝不同，衍射光栅产生的极大明显更尖锐、更明亮，因为来自许多狭缝的相长干涉将光强集中在非常窄的角度区域内。这种锐度使得光栅成为光谱学的理想选择。
2. The Grating Equation | 光栅方程
For a transmission grating illuminated at normal incidence by a plane wavefront, the directions of the principal maxima are given by the grating equation:
d sin θ = n λ
对于垂直入射的平面波前照射的透射光栅，主极大的方向由光栅方程给出：
d sin θ = n λ
Here d is the grating spacing, θ is the angle between the diffracted beam and the normal to the grating, n is the order number (an integer, can be positive, negative or zero), and λ is the wavelength of the light in vacuum.
这里 d 是光栅常数，θ 是衍射光束与光栅法线之间的夹角，n 是级次（整数，可为正、负或零），λ 是光在真空中的波长。
gher orders (n = ±1, ±2, ...) are symmetrically arranged on either side of the zero order.
零级极大（n = 0）出现在 θ = 0° 处，意味着所有波长都会在此直透位置重叠。更高级次（n = ±1, ±2, ...）对称地分布在零级的两侧。
3. Understanding the Variables | 理解各个变量
Grating spacing d: This is the separation between adjacent slits. For a grating with 300 lines per mm, d = (1/300) mm = 3.33 × 10⁻⁶ m. The smaller d is, the larger the angular separation between orders.
光栅常数 d：这是相邻狭缝的间距。对于每毫米 300 条线的光栅，d = (1/300) mm = 3.33 × 10⁻⁶ m。d 越小，各级次之间的角度间隔越大。
Angle θ: orders. Because sin θ ≤ 1, there is a limit on the highest observable order for a given λ.
角度 θ：从法线开始测量。在多数 A-Level 问题中，角度较小，但对于高级次，角度可接近 90°。由于 sin θ ≤ 1，对于给定的波长，所能观察到的最高级次会受到限制。
Order number n: first order on either side. The spectrum is repeated in each order, and the angular dispersion increases with n.
级次 n：n = 0 为中央极大。n = 1 是两侧的第一级。光谱在每个级次中重复出现，并且角色散随 n 的增大而增大。
Wavelength λ: Different wavelengths are diffracted through different angles, creating a spectrum. Shorter wavelengths (blue) are diffracted less than longer wavelengths (red) for the same order.
波长 λ：不同波长会以不同角度衍射，从而形成光谱。在相同级次中，较短波长（蓝光）的衍射角度小于较长波长（红光）。
4. Derivation of the Grating Equation | 光栅方程的推导
Consider two adjacent slits on a transmission grating illuminated by a plane monochromatic wave at normal incidence. Each slit acts as a source of secondary wavelets (Huygens' principle). In a direction at angle θ to the normal, the path difference between wavelets from adjacent slits is d sin θ.
考虑一个透射光栅上两个相邻的狭缝，垂直入射的平面单色波照射在它们之上。每个狭缝作为次波源（惠更斯原理）发出子波。在与法线成 θ 角的方向上，来自相邻狭缝的子波之间的光程差为 d sin θ。
For constructive interference and a bright maximum, this path difference must equal an integer number of wavelengths: d sin θ = n ition must hold for all slit pairs, which makes the intensity add up coherently to form a sharp peak.
要产生相长干涉并形成亮条纹，此光程差必须等于波长的整数倍：d sin θ = n λ，其中 n = 0, ±1, ±2, ...。所有狭缝对都必须满足这一条件，从而使光强相干叠加形成尖锐的峰。
If the light is incident at an angle i to the normal, the more general grating equation becomes d (sin i + sin θ) = n λ. However, the CIE syllabus mainly uses the normal-incidence form.
如果光线以与法线成 i 角的方向入射，更一般的光栅方程变为 d (sin i + sin θ) = n λ。但 CIE 大纲主要涉及垂直入射的形式。
5. Orders of Diffraction and Maximum Possible Orders | 衍射级次与最大可能级次
For a given grating and wavelength, the number of observable orders is limited. Since sin θ cannot exceed 1, the condition n λ / d ≤ 1 gives the maximum order nmax = floor(d/λ). Higher orders may exist mathematically but require sin θ > 1, which is physically impossible.
对于给定的光栅和波长，可观察到的级次数量是有限的。由于 sin θ 不能超过 1，条件 n λ / d ≤ 1 给出了最大级次 nmax = floor(d/λ)。数学上可能存在更高级次，但这要求 sin θ > 1，在物理上是不可能的。
In experiments, partially overlapping orders can occur when different wavelengths coincide at the same angle, e.g. the second-order blue line may overlap with the third-order red line. This overlap is predicted by the grating equation when n1 λ1 = n2 λ2.
在实验中，当不同波长在同一角度重合时，可能会出现级次的部分重叠，例如二级蓝光可能与三级红光重叠。当 n1 λ1 = n2 λ2 时，光栅方程便可预测这种重叠。
6. Angular Dispersion of a Grating | 光栅的角色散
Angular dispersion describes how much the diffraction angle changes with a small change in wavelength. It is mathematically expressed as the derivative dθ/dλ. By differentiating the grating equation with respect to λ (assuming small changes), we obtain:
dθ/dλ = n / (d cos θ)
角色散描述了衍射角如何随波长的微小变化而改变。其数学表达式为导数 dθ/dλ。通过对光栅方程关于 λ 微分（假设微小变化），我们得到：
dθ/dλ = n / (d cos θ)
This shows that angular dispersion increases with order number n and with smaller grating spacing d. It also increases as θ approaches 90° (cos θ small), though at grazing angles the intensity falls. In spectroscopy, higher dispersion means the spectral lines are spread further apart in angle, allowing better separation.
这表明，角色散随级次 n 的增大和光栅常数 d 的减小而增大。当 θ 接近 90° 时角色散也会增大（cos θ 很小），但在掠射角下光强会下降。在光谱学中，较高的角色散意味着谱线在角度上分布得更开，能提供更好的分离度。
7. Resolving Power (R = λ/Δλ = Nn) | 分辨本领
The resolving power R of a diffraction grating is its ability to distinguish between two closely spaced wavelengths, λ and λ + Δλ. It is defined as:
R = λ / Δλ = N n
衍射光栅的分辨本领 R 是其区分两个非常接近的波长 λ 和 λ + Δλ 的能力。它的定义为：
R = λ / Δλ = N n
Here N is the total number of illuminated grating lines, and n is the order of the spectrum. A greater number of lines and the use of higher orders both improve the resolution. For example, a grating with 5000 lines per cm and a width of 2 cm has N = 10,000 lines. In the second order (n = 2), R = 20,000, which can resolve wavelength differences as small as 0.03 nm around λ = 600 nm.
这里 N 是被照明的光栅刻线总数，n 是光谱的级次。刻线数越多、使用的级次越高，都能提高分辨率。例如，一个每厘米 5000 条线、宽度为 2 cm 的光栅，N = 10,000 条线。在第二级（n = 2）下，R = 20,000，能够分辨波长 600 nm 附近小至 0.03 nm 的波长差。
It is important not to confuse resolving power with angular dispersion. Dispersion tells you how far apart two lines are in angle; resolving power tells you whether they can be seen as separate, which depends on the line width produced by the grating.
切勿将分辨本领与角色散混淆。角色散告诉你两条谱线在角度上相距多远；而分辨本领则决定它们是否能够被区分开来，这取决于光栅产生的谱线宽度。
8. Diffraction Grating vs. Double Slit | 衍射光栅与双缝的比较
Although both arrangements rely on interference, diffraction gratings offer several advantages over a double slit. The table below summarises the key differences.
| Property | Double Slit | Diffraction Grating |
|---|---|---|
| Number of slits | 2 | Many (hundreds to thousands per mm) |
| Maxima brightness | Moderate | Very bright and sharp |
| Width of maxima | Broad | Very narrow |
| Equation | d sin θ = nλ (same form) | d sin θ = nλ |
| Spectral resolution | Poor; colours overlap | High; can separate close wavelengths |
| Use in wavelength measurement | Approximate | Precise |
尽管这两种装置都依赖于干涉，但衍射光栅比双缝具有几个优势。上表总结了主要区别。
The sharp maxima of a grating arise because with many slits, destructive interference is nearly complete at angles that only slightly deviate from the exact condition, leading to intense peaks against a dark background.
光栅能产生尖锐的极大，是因为存在众多狭缝，在稍微偏离严格条件的角度上，相消干涉几乎是完全的，从而在暗背景上形成极强的峰。
9. Practical Applications of Diffraction Gratings | 衍射光栅的实际应用
Diffraction gratings are widely used in spectrometers to analyse the composition of light from stars, flames, and chemical samples. In astronomy, grating spectrographs reveal absorption and emission lines that identify elements in stellar atmospheres.
衍射光栅被广泛应用于光谱仪中，用于分析来自恒星、火焰和化学样品的光的成分。在天文学中，光栅摄谱仪会显示出吸收线和发射线，从而识别恒星大气中的元素。
They are also essential in laser technology for selecting specific wavelengths (tuning lasers), in optical communications for wavelength division multiplexing, and in high-resolution monochromators. Reflection gratings are common in compact disc (CD) and DVD players to read digital data encoded as pits on a reflective surface.
它们在激光技术中用于选择特定波长（调谐激光）、在光通信中用于波分复用，以及在高分辨率单色仪中也是必不可少的。反射光栅常用于光盘（CD）和 DVD 播放器中，以读取在反射表面上以凹坑形式编码的数字数据。
The iridescent colours seen on a CD or a butterfly wing arise from natural grating-like structures that diffract white light into its component colours, illustrating the same principle.
在光盘或蝴蝶翅膀上看到的虹彩颜色，就是由于天然的类光栅结构将白光衍射成其组成颜色，这正说明了相同的原理。
10. Measuring Wavelength with a Grating | 用光栅测量波长
A common A-Level experiment uses a transmission grating with a known line spacing to determine the wavelength of a monochromatic source (e.g., a sodium lamp or a laser). The grating is mounted on a spectrometer table, and the collimator produces a parallel beam incident normally on the grating.
常见的 A-Level 实验使用的是已知线间距的透射光栅，来测定单色光源（如钠灯或激光）的波长。光栅安装在分光计台上，准直管产生平行光束垂直入射到光栅上。
The telescope is swung to observe the first-order maximum on both sides. The angular readings for the left and right first-order maxima are recorded, and the difference 2θ1 is obtained. Then θ1 is calculated. Using d sin θ1 = λ, the wavelength can be found.
转动望远镜以观察两侧的一级极大。记录左侧和右侧一级极大的角度读数，并得到差值 2θ1。然后计算出 θ1。再使用 d sin θ1 = λ 便可求出波长。
To improve accuracy, higher orders (if visible) are also measured, and an average value of λ is taken. It is important to ensure that the grating is normal to the incident beam and to avoid parallax errors when reading the vernier scale.
为了提高准确度，也可以测量更高级次（若可见）并取 λ 的平均值。重要的是要确保光栅垂直于入射光束，并在读取游标刻度时避免视差错误。
11. Common Pitfalls and Examiner Tips | 常见陷阱与考试技巧
Unit conversions: Grating spacing d is often given in lines per mm. Always convert to metres: e.g. 300 lines/mm → d = (1/300) × 10⁻³ m = 3.33 × 10⁻⁶ m.
单位换算：光栅常数 d 常常以每毫米的线数给出。务必将其换算为米：例如 300 lines/mm → d = (1/300) × 10⁻³ m = 3.33 × 10⁻⁶ m。
Using sine correctly: θ) when angles are large. A-Level problems often require accurate use of the sine function, so a calculator is essential.
正确使用正弦：许多学生在角度较大时错误地使用 tan θ 或小角度近似（sin θ ≈ θ）。A-Level 题目通常要求精确使用正弦函数，因此必须使用计算器。
Maximum order confusion: Remember that the theoretical maximum order is limited by sin θ = 1. If n λ / d > 1, that order is not visible.
混淆最大级次：记住理论上的最大级次受 sin θ = 1 限制。如果 n λ / d > 1，则该级次不可见。
Zero-order overlap: roduce white light (if the source is white). Do not try to measure dispersion from the zero order.
零级重叠：在 n = 0 处，所有波长重叠并产生白光（如果光源为白光）。不要试图从零级测量色散。
Explaining sharpness: Be ready to explain why grating maxima are sharper than double‑ ence is nearly complete over most angles, except very close to the condition d sin θ = nλ.
解释锐度：需准备好解释为何光栅极大比双缝条纹更尖锐：大量狭缝意味着在绝大多数角度上相消干涉几乎完全，只在非常接近 d sin θ = nλ 的附近才有明显的相长干涉。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图