Interference of Light | 光的干涉

📚 Interference of Light | 光的干涉

Interference is one of the most compelling pieces of evidence for the wave nature of light. When two or more coherent light waves overlap, they combine to form a stable pattern of bright and dark fringes—something that cannot be explained by a particle model alone. In the CIE A-Level Physics syllabus, you are expected to understand the conditions for interference, perform calculations involving fringe spacing, analyse the diffraction grating equation, and explain everyday phenomena such as thin-film colours.

干涉是证明光具有波动性的最有力证据之一。当两束或多束相干光波相遇时，它们会叠加形成明暗相间的稳定条纹图样——这是单纯的微粒模型无法解释的。CIE A-Level 物理大纲要求你掌握干涉产生的条件，会计算条纹间距，能运用衍射光栅方程，并能解释薄膜色彩等日常现象背后的原理。

1. Conditions for Observable Interference | 产生可观察干涉的条件

To produce a steady interference pattern, the overlapping waves must be coherent and monochromatic. Coherence means the waves maintain a constant phase difference over time; in practice, this is achieved by splitting light from a single source into two paths. Monochromatic light (single wavelength) ensures that the fringe spacing is uniform and the pattern does not blur due to overlapping colours.

要产生稳定的干涉图样，叠合的波必须是相干的且为单色光。相干意味着波之间的相位差不随时间变化；实际操作中，通常是将同一光源的光分成两路来实现。单色光（单一波长）能保证条纹间距均匀，不会因不同颜色重叠而使图样变得模糊。

If ordinary white light were used directly without a single slit, the random phase changes in the two separate sources would wash out any interference pattern. This is why Young’s double-slit experiment relies on a narrow single slit placed just before the double slits—it acts as a spatial filter to ensure the waves reaching the two slits originate from the same wavefront.

如果直接使用普通的白光而不先通过单缝，两个独立光源的随机相位变化会把任何干涉图样抹掉。这就是为什么杨氏双缝实验在双缝之前要放置一条窄单缝——它起到空间滤波器的作用，确保到达双缝的波来自同一个波前。

2. Young’s Double-Slit Experiment | 杨氏双缝实验

Young’s experiment is the classic demonstration of two‑source interference. Laser light illuminates two narrow, closely spaced slits, producing an interference pattern of equally spaced bright and dark fringes on a distant screen. Bright fringes occur where the path difference from the two slits is an integer multiple of the wavelength (constructive interference); dark fringes occur where the path difference is an odd multiple of half the wavelength (destructive interference).

杨氏实验是双光源干涉的经典演示。激光照射两条靠得很近的窄缝，在远处的屏上产生等间距的明暗条纹。光程差等于波长整数倍的地方出现亮纹（相长干涉）；光程差等于半波长奇数倍的地方出现暗纹（相消干涉）。

The fringe spacing Δy (distance between the centres of adjacent bright fringes) is given by:

Δy = λD / a

where λ is the wavelength, D is the distance from the slits to the screen, and a is the slit separation. This formula holds as long as D ≫ a and the angles involved are small, which is true for the typical school laboratory setup.

条纹间距 Δy（相邻亮纹中心之间的距离）由下式给出：

Δy = λD / a

其中 λ 是波长，D 是双缝到屏的距离，a 是双缝间距。这个公式在 D ≫ a 且角度很小的条件下成立，这正是典型学校实验室的情况。

If the separation a is halved, the fringe spacing doubles, making the pattern more spread out. If red light (longer λ) is replaced by blue light (shorter λ), the fringes become narrower. The intensity of the bright fringes is roughly uniform across the central region due to the combined effects of single‑slit diffraction and double‑slit interference.

如果缝间距 a 减半，条纹间距就会加倍，图样变得更加展宽。如果把红光（波长较长）换成蓝光（波长较短），条纹会变窄。由于单缝衍射和双缝干涉的共同影响，中央区域的亮纹强度大致均匀。

3. Path Difference and Phase Difference | 光程差与相位差

Interference is governed by the relationship between path difference and phase difference. A path difference Δx corresponds to a phase difference Δφ = (2π/λ)·Δx. Constructive interference occurs when Δx = nλ (n = 0, 1, 2, …), i.e. in phase; destructive interference occurs when Δx = (n + ½)λ, i.e. completely out of phase.

干涉由光程差和相位差之间的关系决定。光程差 Δx 对应的相位差为 Δφ = (2π/λ)·Δx。当 Δx = nλ（n = 0, 1, 2, …），即同相时，发生相长干涉；当 Δx = (n + ½)λ，即完全反相时，发生相消干涉。

For two‑source interference, the path difference at a point on the screen is approximately a sin θ, where θ is the angle subtended at the slits. When θ is small, sin θ ≈ tan θ = y/D, so the condition for the n‑th bright fringe becomes a(y/D) = nλ, leading directly to y = nλD/a and hence Δy = λD/a.

对于双光源干涉，屏幕上某点的光程差近似为 a sin θ，其中 θ 是该点对双缝的张角。当 θ 很小时，sin θ ≈ tan θ = y/D，因此第 n 级亮纹的条件变为 a(y/D) = nλ，直接导出 y = nλD/a，进而得到 Δy = λD/a。

4. The Diffraction Grating | 衍射光栅

A diffraction grating consists of many equally spaced parallel slits (or lines), typically with several hundred lines per millimetre. When monochromatic light falls on a grating, it produces very sharp, well‑separated maxima at angles that satisfy the grating equation:

d sin θ = nλ

where d is the grating spacing (the distance between adjacent slits), n is the order number (0, 1, 2, …), λ is the wavelength, and θ is the angle between the incident beam and the diffracted beam for that order.

衍射光栅由大量等间距的平行狭缝（或刻线）组成，通常每毫米有数百条线。当单色光照射光栅时，会在满足光栅方程的特定角度上产生非常尖锐且分得很开的主极大：

d sin θ = nλ

其中 d 是光栅常数（相邻狭缝的距离），n 是级数（0, 1, 2, …），λ 是波长，θ 是该级衍射光与入射光之间的夹角。

Gratings are preferred over double slits for measuring wavelengths because the maxima are much sharper, allowing more precise angular measurements. The number of slits N determines the sharpness: the angular width of a principal maximum is proportional to 1/N. With a typical grating of 300 lines per millimetre, d = 1/300 mm ≈ 3.33 × 10⁻⁶ m.

测量波长时，光栅优于双缝，因为主极大更加尖锐，角度测量精度更高。狭缝总数 N 决定了锐度：主极大的角宽度正比于 1/N。一个典型的 300 线/毫米光栅，其光栅常数 d = 1/300 mm ≈ 3.33 × 10⁻⁶ m。

When white light is used, the central maximum (n = 0) is white because all wavelengths overlap constructively at θ = 0. Higher orders produce spectra: violet is deviated least and red most, so each order displays a continuous spectrum, but orders may overlap for higher n.

使用白光时，零级主极大（n = 0）为白色，因为所有波长的光在 θ = 0 处都发生相长干涉。更高级次则形成光谱：紫光偏转最小，红光偏转最大，因此每一级都呈现连续光谱，但级次较高时可能出现重叠。

5. Deriving the Grating Formula | 光栅方程的推导

The derivation of d sin θ = nλ relies on considering parallel rays from adjacent slits. For two neighbouring slits, the extra distance travelled by the lower ray is d sin θ. When this path difference equals a whole number of wavelengths, the waves from all slits arrive in phase and reinforce each other, giving a principal maximum.

d sin θ = nλ 的推导基于考虑相邻狭缝的平行光线。对两条相邻狭缝，下方光线多走的路程为 d sin θ。当这个光程差等于波长的整数倍时，所有狭缝的波都同相到达，彼此加强，形成主极大。

If we consider three slits, cancellation also occurs at angles between the principal maxima, leading to subsidiary maxima and minima. As the number of slits increases, the intensity of the principal maxima grows while the subsidiary maxima become smaller and the minima fill in the gaps, explaining why grating maxima are sharp.

如果考虑三条狭缝，在各级主极大之间的角度上也会出现相消，产生次极大和极小。随着狭缝数目增加，主极大的强度增大，而次极大变得越来越小，极小填补了间隙，这就解释了为什么光栅的主极大非常尖锐。

6. Thin‑Film Interference | 薄膜干涉

Thin‑film interference arises when light reflects off the top and bottom surfaces of a thin transparent film (oil slick, soap bubble, anti‑reflection coating). The two reflected waves superpose; whether they interfere constructively or destructively depends on the film thickness t, the wavelength λ, the refractive index n of the film, and any phase changes upon reflection.

薄膜干涉发生在光从透明薄膜（油膜、肥皂泡、增透膜）的上下表面反射时。两束反射光叠加；发生相长还是相消取决于膜的厚度 t、波长 λ、膜的折射率 n 以及反射时可能存在的相位跃变。

An important phase‑change rule applies: when light reflects off a medium of higher refractive index, it undergoes a phase change of π (equivalent to an extra path of λ/2). If light reflects off a medium of lower refractive index, there is no phase change. This rule must be combined with the optical path length 2nt (for near‑normal incidence) to write the correct constructive or destructive condition.

一个重要的相位变化规则：光从折射率较高的介质反射时，会发生 π 的相位跃变（相当于额外增加 λ/2 的光程）；从折射率较低的介质反射时不发生相位变化。在近垂直入射时，必须结合光程 2nt，并根据具体边界条件写出正确的相长或相消条件。

For an oil film (n₁ for air, n₂ for oil, n₃ for water with n₂ > n₁ and n₂ > n₃): both reflections involve a phase change of π—the first at air–oil because n₂ > n₁, the second at oil–water because n₂ > n₃. As a result, the net phase difference due to reflections is zero. Constructive interference for reflected light then occurs when 2n₂t = mλ (m = 0, 1, 2, …), giving bright colours for those wavelengths in white light.

对于油膜（空气折射率 n₁，油折射率 n₂，水折射率 n₃，且 n₂ > n₁, n₂ > n₃）：两次反射都发生 π 的相位跃变——第一次在空气-油界面因为 n₂ > n₁，第二次在油-水界面因为 n₂ > n₃。因此，反射引起的净相位差为零。反射光发生相长干涉的条件变为 2n₂t = mλ（m = 0, 1, 2, …），白光中满足该条件的波长会呈现出鲜明的颜色。

7. Anti‑Reflection Coatings | 增透膜

Anti‑reflection coatings on lenses use thin‑film interference to cancel reflected light. A coating of material with refractive index n_c is deposited on glass (n_g), with n_air < n_c < n_g. Both reflections undergo a π phase change (air–coating and coating–glass), so the net phase difference from reflections is again zero. The condition for destructive interference in the reflected beam is 2n_c t = (m + ½)λ, where the coating thickness t is typically chosen for m = 0, i.e. t = λ/(4n_c) for the central wavelength of the visible spectrum.

镜头上的增透膜利用薄膜干涉来消除反射光。在玻璃（n_g）上镀一层折射率为 n_c 的材料，满足 n_空气 < n_c < n_g。两次反射都发生 π 相位跃变（空气-膜层和膜层-玻璃），反射引起的净相位差再次为零。反射光发生相消干涉的条件是 2n_c t = (m + ½)λ，通常选择 m = 0，即对可见光谱的中心波长，膜厚取 t = λ/(4n_c)。

Such coatings greatly reduce glare and increase the transmission of light through multi‑element camera lenses. The residual faint purple or green tint often seen on coated lenses comes from the fact that the exact λ/(4n_c) condition holds for only one wavelength; other wavelengths are partially reflected.

这种镀膜大幅减少了眩光，提高了多镜片相机镜头的透光率。镀膜镜头表面常见的淡紫色或淡绿色反光，正是由于精确的 λ/(4n_c) 条件只对某一波长完全成立，其他波长的光仍被部分反射所致。

8. Fringe Visibility and Coherence Length | 条纹可见度与相干长度

In Young’s experiment, if the path difference between the two beams becomes too large, the fringes fade. This is because real light sources do not emit infinitely long wave trains; they emit in short bursts. The coherence length is the maximum path difference over which interference remains observable, and it is related to the monochromaticity of the source. A laser, with its very narrow bandwidth, has a long coherence length, whereas white light has a very short coherence length.

在杨氏实验中，如果两束光的光程差过大，条纹就会消失。这是因为实际光源不会发出无限长的波列，而是以短脉冲形式发光。相干长度是仍能观察到干涉现象的最大光程差，它与光源的单色性好坏直接相关。激光由于带宽极窄，相干长度很长；而白光的相干长度非常短。

This is why, when using a discharge lamp with a filter to approximate monochromatic light, you must ensure that the path difference in your setup remains well below the coherence length, otherwise the fringe pattern will be of low contrast.

这就解释了为什么使用加了滤光片的放电灯来获取近似单色光时，必须确保光路中的光程差远小于相干长度，否则干涉条纹的对比度会很差。

9. Measuring the Wavelength of Light | 测量光的波长

A diffraction grating provides one of the most accurate classroom methods for determining the wavelength of a laser or spectral line. You measure the angle θ for several orders n (both left and right of the central maximum) and plot a graph of sin θ against n. The gradient of the straight line is λ/d, from which λ can be calculated since d is known from the grating’s ruling density.

衍射光栅是课堂上最精确地测定激光或光谱线波长的方法之一。测量各级 n（零级主极大左右两侧）的衍射角 θ，然后绘制 sin θ 对 n 的图像。所得直线的斜率为 λ/d，由于光栅常数 d 已知（由光栅刻线密度得出），即可求出 λ。

Using a grating reduces uncertainty compared with Young’s double slits because the angles are larger and the maxima are sharper. However, care must be taken to align the grating perpendicular to the incident beam and to measure angles symmetrically to cancel zero‑error in the spectrometer.

与杨氏双缝相比，使用光栅可降低不确定度，因为衍射角更大、主极大更尖锐。但需要注意将光栅调至与人射光垂直，并通过对称测量左右两侧的角度来消除分光计中的零点误差。

10. Common Misconceptions and Pitfalls | 常见误解与易错点

Do not confuse separation a with slit width. In double‑slit interference, the slit width affects the single‑slit diffraction envelope, which modulates the interference fringes, but the fringe spacing Δy depends only on a, D and λ.

不要混淆缝间距 a 与缝宽。在双缝干涉中，缝宽影响的是单缝衍射包络，它会调制干涉条纹的强度，但条纹间距 Δy 只取决于 a、D 和 λ。
Interference and diffraction are both wave phenomena, and they are not mutually exclusive. A diffraction grating produces an interference pattern, but we call it a grating ‘diffraction’ pattern because the mathematics arises from the superposition of many diffracted waves.

干涉和衍射都是波动现象，二者并不互斥。衍射光栅产生的是干涉图样，但习惯上称之为光栅“衍射”图样，因为其数学描述来自许多衍射波的叠加。
When using Δy = λD/a, remember that D is the perpendicular distance from the plane of the slits to the screen. If the screen is angled, the geometry becomes more complicated and the simple formula no longer applies.

使用 Δy = λD/a 时，注意 D 是双缝所在平面到屏幕的垂直距离。如果屏幕倾斜，几何关系变复杂，上述简易公式就不再适用。
In thin‑film problems, always sketch the boundaries and note the refractive indices to decide on phase changes before writing the path condition.

在薄膜干涉问题中，一定要先画边界并标注折射率，判断好相位跃变情况后，再写出光程条件。

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Interference of Light | 光的干涉