Diffraction Gratings | 衍射光栅

📚 Diffraction Gratings | 衍射光栅

Diffraction gratings are essential optical components that split light into its component wavelengths by using a large number of equally spaced parallel slits. In A-Level Physics, the grating equation d sin θ = n λ is a key tool for predicting the directions of constructive interference and for measuring wavelength.

衍射光栅是一种重要的光学元件，它利用大量等间距平行狭缝将光按波长分开。在 A-Level 物理中，光栅方程 d sin θ = n λ 是预测相干加强方向以及测量波长的关键工具。

1. What is a Diffraction Grating? | 什么是衍射光栅？

A diffraction grating consists of many closely spaced, identical parallel slits or rulings. Transmission gratings transmit light through the slits, while reflection gratings reflect light from a grooved surface.

衍射光栅由许多紧密排列、完全相同的平行狭缝或刻线组成。透射光栅让光透过狭缝，反射光栅则从刻槽表面反射光。

The slit spacing d is the distance between adjacent slits. It is often quoted as the number of lines per millimetre N, and the two are related by:

狭缝间距 d 是相邻狭缝之间的距离，通常以每毫米线数 N 来表示，两者之间的关系为：

d = 1 ÷ N

For example, a grating with 600 lines per mm has d = 1 ÷ (600 × 10³ m⁻¹) = 1.67 × 10⁻⁶ m.

例如，每毫米 600 条线的光栅，其 d = 1 ÷ (600 × 10³ m⁻¹) = 1.67 × 10⁻⁶ m。

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2. How a Grating Forms Maxima | 光栅如何形成极大

When monochromatic light passes through a grating, each slit acts as a coherent secondary source. Waves from adjacent slits travel different distances to reach a distant screen or detector.

当单色光通过光栅时，每条狭缝都相当于一个相干次光源。来自相邻狭缝的波到达远处屏幕或探测器时经过的路程不同。

The path difference between waves from neighbouring slits is d sin θ, where θ is the angle measured from the normal to the grating.

相邻狭缝波之间的光程差为 d sin θ，其中 θ 是从光栅法线方向量起的角度。

Constructive interference occurs when this path difference is an integer multiple of the wavelength. This condition produces the bright principal maxima.

当该光程差等于波长的整数倍时，发生相长干涉，从而产生明亮的主极大。

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3. The Grating Equation | 光栅方程

For principal maxima, the path difference must equal n λ, where n is the order number given by n = 0, ±1, ±2, … This gives the grating equation:

对于主极大，光程差必须等于 n λ，其中 n 是级数，取值为 n = 0, ±1, ±2, …。由此得到光栅方程：

d sin θ = n λ

In this equation, d is the slit spacing, θ is the angle of the nth-order maximum from the normal, and λ is the wavelength of the incident light.

在此方程中，d 是狭缝间距，θ 是第 n 级极大与法线的夹角，λ 是入射光的波长。

The zero order n = 0 always occurs at θ = 0, meaning all wavelengths pass straight through undeviated.

零级 n = 0 总是出现在 θ = 0 处，这意味着所有波长都直线通过而不发生偏转。

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4. Orders of Diffraction | 衍射级

The central bright line is the zero order. On either side of it, the first-order maxima appear at angles satisfying sin θ = λ / d, the second-order maxima satisfy sin θ = 2 λ / d, and so on.

中央亮线是零级。在它两侧，一级极大出现在满足 sin θ = λ / d 的角度处，二级极大满足 sin θ = 2 λ / d，依此类推。

Positive and negative orders are symmetric about the centre, so each wavelength produces a pair of first-order spots, a pair of second-order spots, and so on.

正级和负级关于中心对称，因此每个波长都会产生一对一级亮点、一对二级亮点，依此类推。

The maximum possible order is limited because sin θ cannot exceed 1. Therefore:

可观察到的最大级数受到 sin θ 不能超过 1 的限制，因此：

n ≤ d ÷ λ

If d is smaller than λ, only the zero order is observable.

如果 d 小于 λ，则只能观察到零级。

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5. Sharpness and Intensity of Maxima | 极大的锐度与强度

A grating produces much sharper and brighter maxima than a double slit. With many slits, destructive interference occurs for even small deviations from the exact angle, so the bright lines are narrow and the dark regions are wide.

光栅产生的极大比双缝干涉更锐利、更明亮。由于狭缝数目很多，只要稍微偏离准确角度就会发生相消干涉，因此亮线很窄，暗区很宽。

The intensity of principal maxima is proportional to N², where N is the number of illuminated slits. This makes grating spectra highly suitable for precise wavelength measurements.

主极大的强度与受照狭缝数 N 的平方成正比，即 N²。这使得光栅光谱非常适合精确测量波长。

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6. Diffraction Grating vs Double Slit | 衍射光栅与双缝干涉对比

The table below summarises the key differences between a double slit and a diffraction grating.

下表总结了双缝与衍射光栅之间的主要区别。

Property | 性质	Double slit | 双缝	Diffraction grating | 光栅
Number of slits | 狭缝数	Two | 两条	Many, often 300–600 per mm | 很多，通常每毫米 300–600 条
Maxima | 极大	Broad and moderately bright | 较宽且亮度中等	Very sharp and intense | 非常锐利且强度高
Dark regions | 暗区	Narrow | 较窄	Wide and well defined | 宽阔且边界清晰
Best use | 最佳用途	Demonstrating basic interference | 演示基本干涉	Precise wavelength measurement | 精确测量波长

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7. White Light and Spectra | 白光与光谱

With white light, the zero order is a central white line because all wavelengths overlap at θ = 0. For n ≠ 0, each wavelength diffracts at a different angle, producing a continuous spectrum.

使用白光时，零级是中央白色亮线，因为所有波长在 θ = 0 处重叠。对于 n ≠ 0，各波长以不同角度衍射，形成连续光谱。

Violet light has the shortest visible wavelength and diffracts least, so it appears closest to the centre. Red light has the longest visible wavelength and appears farthest from the centre.

紫光在可见光中波长最短，衍射角度最小，因此最靠近中心。红光在可见光中波长最长，离中心最远。

Each order produces its own spectrum. Higher orders are more spread out but dimmer, and they may overlap with neighbouring orders.

每一级都会产生自身的光谱。级越高谱线展开越大，但亮度越暗，并且可能与其他级次发生重叠。

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8. Overlapping Orders | 级次重叠

Overlap occurs when different orders and wavelengths satisfy the same angle. This happens when:

当不同级次和波长满足同一角度时，就会发生重叠。其条件为：

n₁ λ₁ = n₂ λ₂

For example, the second-order line of 400 nm light can coincide with the first-order line of 800 nm light if both wavelengths are present in the incident beam.

例如，若入射光束中同时存在 400 nm 和 800 nm 的光，则 400 nm 光的二级谱线可能与 800 nm 光的一级谱线重合。

This overlap is important in spectroscopy. Filters or a pre-dispersing element may be needed to separate overlapping orders before detection.

这种重叠在光谱学中非常重要。在检测之前，可能需要使用滤光片或前置色散元件来区分重叠的级次。

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9. Experimental Determination of Wavelength | 实验测定波长

A common CIE experiment uses a laser or a collimated lamp, a grating of known spacing, and a screen. Measure the distance y from the central maximum to the nth-order spot, and the distance D from the grating to the screen.

CIE 常见实验使用激光或准直光源、已知间距的光栅和屏幕。测量中央极大到第 n 级亮点的距离 y，以及光栅到屏幕的距离 D。

The angle θ is found using:

角度 θ 可通过下式求得：

tan θ = y ÷ D

Then the wavelength is calculated from the grating equation λ = d sin θ ÷ n.

然后根据光栅方程 λ = d sin θ ÷ n 计算波长。

For better accuracy, use a spectrometer to measure θ directly and take readings on both sides of the zero order to average out any misalignment of the grating.

为提高精度，可以使用分光计直接测量 θ，并在零级两侧分别读数，取平均值以消除光栅未对准造成的误差。

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10. Angular Dispersion and Resolving Power | 角色散与分辨本领

Angular dispersion describes how rapidly the diffraction angle changes with wavelength. It is greater for higher orders and for gratings with smaller slit spacing.

角色散描述衍射角随波长变化的快慢。级次越高、狭缝间距越小，角色散越大。

From the grating equation, the angular dispersion is approximately:

由光栅方程可得角色散近似为：

Δθ ÷ Δλ ≈ n ÷ (d cos θ)

Resolving power measures the ability to separate two close wavelengths. It is given by R = λ ÷ Δλ = nN, where N is the total number of illuminated lines.

分辨本领衡量区分两条接近波长的能力，其定义为 R = λ ÷ Δλ = nN，其中 N 是受照线条总数。

A grating with many illuminated lines can resolve closely spaced lines such as the sodium doublet D lines.

具有大量受照刻线的光栅可以分辨间距很近的谱线，例如钠双黄线。

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11. Applications of Diffraction Gratings | 衍射光栅的应用

Diffraction gratings are used in spectrometers, astronomy, telecommunications, and laser systems. They allow scientists to identify elements from emission or absorption spectra and to tune wavelengths in optical devices.

衍射光栅广泛应用于光谱仪、天文学、电信和激光系统。科学家利用它们根据发射光谱或吸收光谱识别元素，并在光学器件中调节波长。

Reflection gratings are common in research instruments, while inexpensive transmission gratings are used in classroom demonstrations and handheld spectroscopes.

反射光栅常用于科研仪器；廉价的透射光栅则用于课堂演示和手持式分光镜。

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12. Worked Example | 例题

A grating has 500 lines per mm. Monochromatic light of wavelength 650 nm is incident normally. Calculate the angle of the second-order maximum and the highest observable order.

一个光栅每毫米有 500 条线，波长为 650 nm 的单色光垂直入射。计算二级极大的角度以及可观察到的最大级次。

Step 1: Find the slit spacing d.

步骤 1：求狭缝间距 d。

d = 1 × 10⁻³ m ÷ 500 = 2.00 × 10⁻⁶ m

Step 2: For the second-order maximum, use n = 2.

步骤 2：对于二级极大，取 n = 2。

sin θ = 2 × 650 × 10⁻⁹ m ÷ 2.00 × 10⁻⁶ m = 0.650

θ = 40.5°

Step 3: The highest observable order is the largest integer n for which sin θ ≤ 1.

步骤 3：可观察到的最大级次是满足 sin θ ≤ 1 的最大整数

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