📚 Interference | 干涉
Interference is a fundamental phenomenon of waves in which two or more coherent waves superpose to form a resultant wave of greater, lower, or the same amplitude. The principle of superposition governs this effect, giving rise to alternating regions of constructive and destructive interference. Understanding interference is essential for explaining the wave nature of light and forms the basis of many optical instruments and precision measurement techniques.
干涉是波的一种基本现象，两列或多列相干波叠加形成一列振幅更大、更小或相同的合波。叠加原理支配着这一效应，产生明暗交替的相长干涉和相消干涉区域。理解干涉对于解释光的波动性至关重要，也是许多光学仪器和精密测量技术的基础。
1. Conditions for Interference | 干涉发生的条件
For a stable, observable interference pattern, the overlapping waves must be coherent. Coherence implies that the waves maintain a constant phase difference over time. In practice, this requires the sources to emit waves of the same frequency and with a fixed phase relationship. If the phase difference varies randomly, the interference fringes will shift rapidly and blend into a uniform average intensity.
为了获得稳定、可观察的干涉图样，叠加的波必须是相干的。相干意味着波之间随时间保持恒定的相位差。实际中，这要求波源发射相同频率的波，并且具有固定的相位关系。若相位差随机变化，干涉条纹将快速移动并融合成均匀的平均强度。
Moreover, the amplitudes of the interfering waves should be comparable. When one wave dominates, the contrast between bright and dark fringes diminishes, making the pattern difficult to distinguish. For the clearest fringes, sources of nearly equal amplitude are preferred.
此外，干涉波的振幅应当相近。当一列波占主导时，明暗条纹的对比度降低，图样难以分辨。为获得最清晰的条纹，最好使用振幅近乎相等的波源。
2. Young's Double-Slit Experiment | 杨氏双缝实验
Young's double-slit experiment provides classic evidence for the wave nature of light. Monochromatic light illuminates two narrow, closely spaced parallel slits, S₁ and S₂. These slits act as two coherent point sources emitting spherical wavefronts in phase. A single narrow slit is often placed before the double slit to ensure that the light reaching the two slits comes from the same wavefront, guaranteeing spatial coherence.
杨氏双缝实验为光的波动性提供了经典证据。单色光照射两条狭窄、相距很近的平行狭缝 S₁ 和 S₂。这两个狭缝相当于两个同相位发出球面波前的相干点光源。通常在双缝前放置一个单缝，以确保抵达双缝的光来自同一波前，从而保证空间相干性。
The waves diffract from the slits and overlap on a distant screen. At the central point of the screen, the optical paths are equal, producing a bright fringe. Moving away from the centre, alternating dark and bright fringes appear as the path difference increases. The fringe pattern is symmetrical and equally spaced for small angles.
波从狭缝衍射出来，在远处的屏幕上重叠。在屏幕中央点，两束光的光程相等，形成亮纹。偏离中心，随着光程差的增大，出现交替的暗纹和亮纹。在小角度条件下，条纹图样对称且等间距分布。
3. Fringe Spacing Formula | 条纹间距公式
For Young's experiment with slit separation d, screen distance D (where D is much larger than d), and wavelength λ, the fringe spacing (distance between adjacent bright or dark fringes) is given by:
对于缝间距为 d、双缝到屏幕距离为 D（且 D 远大于 d）、波长为 λ 的杨氏实验，条纹间距（相邻亮纹或暗纹之间的距离）由下式给出：
Δx = λD / d
θ ≈ y/D, we obtain yₘ = mλD/d. The spacing between successive fringes is Δx = λD/d, which is uniform across the pattern.
该关系式基于小角度近似：第 m 级亮纹的角位置满足 d sinθ = mλ，又因 sinθ ≈ tanθ ≈ y/D，可得 yₘ = mλD/d。相邻条纹的间距为 Δx = λD/d，在整个图样中是均匀的。
From this equation, longer wavelengths produce wider fringe spacing, shorter wavelengths narrower spacing. Thus, red light fringes are more widely separated than blue light fringes, a key point when discussing white-light interference.
由该公式可知，波长越长条纹间距越大，波长越短间距越小。因此，红光的条纹间距大于蓝光，在讨论白光干涉时这是一个关键点。
4. Path Difference and Phase Difference | 光程差与相位差
The nature of interference at a point depends on the optical path difference (OPD) between the waves arriving from the two sources. In a medium of refractive index n, OPD equals n times the geometrical path difference. The corresponding phase difference Δφ is:
某点的干涉性质取决于从两个源抵达该点的波的光程差。在折射率为 n 的介质中，光程差等于几何路程差乘以 n。对应的相位差 Δφ 为：
Δφ = (2π / λ) × (optical path difference)
Constructive interference (bright fringe) occurs when the optical path difference is an integer multiple of the wavelength:
当光程差等于波长的整数倍时发生相长干涉（亮纹）：
OPD = mλ, m = 0, 1, 2, ...
Destructive interference (dark fringe) occurs for an odd multiple of half a wavelength:
当光程差等于半波长的奇数倍时发生相消干涉（暗纹）：
OPD = (m + ½)λ, m = 0, 1, 2, ...
In Young' n ≈ 1 in air, the conditions become d sinθ = mλ for maxima and d sinθ = (m+½)λ for minima.
在杨氏实验中，几何路程差近似为 d sinθ，空气中 n ≈ 1，因此光程差条件变为：极大值处 d sinθ = mλ，极小值处 d sinθ = (m+½)λ。
5. Coherence and Light Sources | 相干性与光源
A practical interference setup demands a highly coherent light source. Lasers are ideal because they produce monochromatic light with excellent temporal and spatial coherence, yielding sharp, high-contrast fringes. Before lasers, a narrow single slit served to extract a spatially coherent portion from a broad thermal source, ensuring the two slits were illuminated by the same wavefront section.
实用的干涉装置需要高相干度的光源。激光是理想选择，因为它能产生单色性极好的光，并具有优异的时空相干性，可得到清晰、高对比度的条纹。在激光出现之前，人们通过一个窄单缝从广谱热光源中提取空间相干部分，确保双缝由同一波前区域照明。
Temporal coherence refers to the correlation of the wave at different times; it is related to the monochromaticity (bandwidth) of the source. A long coherence length means interference can be observed even for large path differences. White light has a very short coherence length because it contains many wavelengths, so interference fringes are seen only near the centre, where the path difference is close to zero.
时间相干性指波在不同时刻的相关性，与光源的单色性（带宽）有关。相干长度长意味着即使光程差较大，仍能观察到干涉。白光包含许多波长，相干长度很短，因此干涉条纹只在中心附近光程差接近于零的区域可见。
6. Interference in Thin Films | 薄膜干涉
Vivid colours observed on soap bubbles and oil slicks on water are due to interference caused by thin films. A thin film of thickness t and refractive index n is illuminated by light. Part of the incident light reflects off the top surface, and part transmits, reflects off the bottom surface, and then exits back into the original medium. These two reflected rays have travelled different optical paths and can interfere when collected by the eye or a lens.
肥皂泡和水面油膜上看到的鲜艳色彩来源于薄膜干涉。厚度为 t、折射率为 n 的薄膜受到光照。一部分入射光在上表面反射，另一部分透射后在下表面反射，再回到原介质。这两束反射光经历了不同的光程
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