Diffraction from a Uniformly Radiating Strip | 均匀辐射带的衍射

📚 Diffraction from a Uniformly Radiating Strip | 均匀辐射带的衍射

Diffraction is a cornerstone of wave physics that arises whenever a wave encounters an obstacle or aperture. In this article, we examine the diffraction pattern generated by a uniformly radiating strip, a problem that elegantly unites calculus, wave superposition, and phasor diagrams. For IB Mathematics students, especially those exploring the Higher Level Calculus option or the applications of integration in physics, this derivation provides a rich context for definite integrals, complex numbers, and trigonometric limits. It also supports the mathematical rigour demanded by the IB Physics syllabus.

衍射是波动物理学的基石，每当波遇到障碍物或孔径时就会发生。本文研究均匀辐射带产生的衍射图样，这一问题巧妙地将微积分、波的叠加和相量图结合在一起。对于IB数学学生，尤其是那些探索高级微积分选项或物理中积分应用的学生，这一推导为定积分、复数和三角极限提供了丰富的背景。它也支撑了IB物理教学大纲所要求的数学严谨性。

1. The Physical Model and Assumptions | 物理模型与假设

We consider a long, straight strip of width a that emits monochromatic radiation uniformly and in phase across its entire surface. This situation is analogous to a single slit in optics, but treated as a continuous array of point sources. The observation screen is placed far enough away that the Fraunhofer (far-field) diffraction condition applies, meaning that all wavelets reaching a given point on the screen can be treated as parallel. The strip is illuminated normally, or self-radiating, ensuring zero initial phase difference across its width.

我们考虑一个宽度为a的长直条带，它在其整个表面上均匀且同相地发射单色辐射。这种情况类似于光学中的单缝，但被视为连续的点源阵列。观察屏放置得足够远，满足夫琅禾费（远场）衍射条件，这意味着到达屏幕上某一点的所有子波都可以视为平行的。条带被正入射照射或自身辐射，确保在其宽度上初始相位差为零。

nfinitesimal segment dx at coordinate x contributes an electric field wavelet with amplitude proportional to dx. All such segments radiate in unison, and we seek the resultant field at a distant point P located at an angle θ measured from the normal to the strip.

选择坐标系，使条带沿水平轴从x = -a/2 延伸到 x = a/2。位于坐标x处的无穷小线段dx贡献一个振幅与dx成正比的电场子波。所有这些线段同步辐射，我们寻找位于与条带法线成θ角的远处点P处的合成场。

2. Path Difference and the Phase Factor | 路径差与相位因子

the wavelength. Hence the phase difference for the element at x is δ(x) = (2π/λ) x sin θ.

为了到达点P，来自位置x的子波必须比来自中心x = 0的子波多走一段额外的距离。根据几何关系，这一路径差为Δ = x sin θ。就相位而言，路径差Δ对应相位移动δ = (2π/λ) Δ，其中λ是波长。因此，位于x的线段的相位差为δ(x) = (2π/λ) x sin θ。

A positive x means the wavelet has to travel further, so it lags behind the central wavelet, corresponding to a phase factor e^{i k x sin θ} where ε₀ is the amplitude per unit width. For time-averaged intensity, the common factor e^iωt cancels, leaving the spatially dependent phasor.

正的x意味着子波必须传播更远的距离，因此它落后于中心子波，对应相位因子e^{i k x sin θ}，其中k = 2π/λ。使用复指数表示，来自线段dx在P点的贡献为dE = ε₀ dx · e^(i(ωt - k x sin θ))，其中ε₀是单位宽度的振幅。对于时间平均强度，公共因子e^iωt可以约去，留下依赖于空间的相量。

3. Setting Up the Resultant Electric Field Integral | 建立合成电场积分

The total complex electric field at angle θ is the coherent sum (integral) of all infinitesimal contributions across the strip. Since the strip radiates uniformly, the amplitude ε₀ is constant, and we integrate over the strip's width:

E(θ) = ε₀ ∫₍₋ₐ/₂₎^(a/2) e^{i k x sin θ} dx

在角度θ处的总复数电场是所有无穷小贡献的相干叠加（积分）。由于条带均匀辐射，振幅ε₀是常数，我们对条带宽度进行积分：

E(θ) = ε₀ ∫₍₋ₐ/₂₎^(a/2) e^{i k x sin θ} dx

This integral has a standard form: ∫ e^iαx dx = (1/(iα)) e^iαx e evaluate it between the symmetric limits. The symmetry simplifies the result using Euler's relation sin β = (e^iβ - e^-iβ)/(2i).

这个积分具有标准形式：∫ e^iαx dx = (1/(iα)) e^iαx。令α = k sin θ，我们在对称的上下限之间计算它。对称性利用欧拉关系sin β = (e^iβ - e^-iβ)/(2i)简化结果。

Substituting the limits -a/2 and a/2 gives the phasor difference in the numerator, leading to an expression containing sin(ka/2 sin θ). The total amplitude then emerges as a sinc function.

代入上下限-a/2和a/2，分子产生相量差，得出包含sin(ka/2 sin θ)的表达式。总振幅随后呈现为sinc函数。

4. Evaluating the Integral Step by Step | 逐步计算积分

Let β = (k a sin θ)/2 = (π a sin θ)/λ. After integration, the field becomes:

E(θ) = ε₀ [ (e^iβ - e^-iβ) / (i k sin θ) ]

Using 2i sin β = e^iβ - e^-iβ, we obtain:

E(θ) = ε₀ a [ sin β / β ]

令β = (k a sin θ)/2 = (π a sin θ)/λ。积分后，电场变为上述形式。利用2i sin β = e^iβ - e^-iβ，我们得到E(θ) = ε₀ a [ sin β / β ]。

β/β → 1). The ratio sin β/β is known as the sinc function, which oscillates and decays as |β| increases. Its square gives the relative intensity distribution on the screen.

因子ε₀ a表示最大可能振幅（当θ = 0时，sin β/β → 1）。比值sin β/β称为sinc函数，它随着|β|增大而振荡并衰减。它的平方给出屏幕上相对强度分布。

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5. The Intensity Distribution Formula | 强度分布公式

Since the intensity I is proportional to the square of the field amplitude, we write:

I(θ) = I₀ [ sin( (π a sin θ)/λ ) / ( (π a sin θ)/λ ) ]²

其中I₀是中心最大强度。这是一维夫琅禾费衍射图样的特征性sinc²分布。

he function decays rapidly; by β = π, the first minimum is reached, giving zero intensity.

在θ = 0处，我们应用标准极限lim_(u→0) sin u / u = 1，确认中心极大具有强度I₀。该函数迅速衰减；到β = π时，达到第一个极小，强度为零。

6. Minima and Their Physical Meaning | 极小值及其物理意义

r any non-zero integer m. Substituting the definition of β gives the famous single-slit diffraction minima condition:

a sin θ = m λ , m = ±1, ±2, ±3, …

极小值出现在sin β = 0但β ≠ 0时。当β = mπ时，对任何非零整数m满足条件。代入β的定义，得到著名的单缝衍射极小条件：a sin θ = m λ , m = ±1, ±2, ±3, …

t direction. The first minimum at m = 1 defines the boundary of the central bright fringe.

物理上，这意味着当条带两边缘之间的路径差等于λ的整数倍时，成对到达的子波在该方向上完全相互抵消。m = 1的第一个极小定义了中央亮纹的边界。

7. Secondary Maxima and Their Intensities | 次级极大及其强度

= 1, 2, 3,... The approximate positions are β₁ ≈ 1.43π, β₂ ≈ 2.46π, β₃ ≈ 3.47π, and so on.

在相邻极小值之间，存在次级极大值，其强度非零但远小于中心峰。这些近似出现在β满足tan β = β时，图形上对应于β值略小于(m + 1/2)π，m = 1, 2, 3,... 近似位置是β₁ ≈ 1.43π, β₂ ≈ 2.46π, β₃ ≈ 3.47π，依此类推。

n by Iₘ/I₀ = [sin βₘ / βₘ]². Numerically, this gives: first secondary max ≈ 0.0472 I₀ (4.7%), second ≈ 0.0165 I₀ (1.65%), third ≈ 0.0083 I₀ (0.83% pid drop in brightness explains why only a few fringes are visible in most demonstrations.

这些次级峰的相对强度由Iₘ/I₀ = [sin βₘ / βₘ]²给出。数值上：第一个次级极大约为0.0472 I₀（4.7%），第二个约为0.0165 I₀（1.65%），第三个约为0.0083 I₀（0.83%）。亮度的迅速下降解释了为什么在大多数演示中只能看到少数条纹。

8. Angular Width of the Central Maximum | 中央极大的角宽度

Δθ_central ≈ 2λ/a. This inversely proportional relationship is a hallmark of diffraction.

中央极大的角半宽度由两侧的第一个极小值决定。对于m = ±1，条件a sin θ₁ = λ给出sin θ₁ = λ/a。在小角度近似下（λ << a），这简化为θ₁ ≈ λ/a（弧度）。因此，中央亮纹的角全宽为Δθ_central ≈ 2λ/a。这种反比关系是衍射的标志。

bout 0.63°. This small angle justifies the paraxial approximation used in many optical setups.

例如，对于λ = 550 nm的绿光，条带宽度a = 0.10 mm，角全宽为Δθ ≈ 2 × (550×10⁻⁹ m) / (0.10×10⁻³ m) = 1.10×10⁻² 弧度，约0.63°。这个小角度证明了在许多光学设置中使用的近轴近似。

9. Connection to Fourier Transforms | 与傅里叶变换的联系

From a higher mathematical perspective, the integral ∫₍₋ₐ/₂₎^(a/2) e^{i k x sin θ} dx is precisely the spatial Fourier transform of the aperture function. The strip can be represented as a rectangular function rect(x/a), and its far-field diffraction pattern is the squared modulus of its Fourier transform. This powerful concept allows students to extend the analysis to more complex aperture shapes, such as circular apertures (producing Airy disks) or multiple slits.

从更高的数学角度来看，积分∫₍₋ₐ/₂₎^(a/2) e^{i k x sin θ} dx正是孔径函数的空间傅里叶变换。条带可以表示为矩形函数rect(x/a)，其远场衍射图样是其傅里叶变换的模平方。这一强大的概念使学生能够将分析扩展到更复杂的孔径形状，如圆形孔径（产生艾里斑）或多缝。

In IB Mathematics HL, the study of Fourier series and transforms (depending on the option) links directly to this kind of physics problem. The variable (sin θ)/λ acts as a spatial frequency. Thus, narrower strips correspond to broader diffraction patterns, reflecting the reciprocal spreading inherent in Fourier analysis—a concept analogous to the Heisenberg uncertainty principle.

在IB数学HL中，傅里叶级数和变换的学习（取决于选项）与此类物理问题直接相关。变量(sin θ)/λ充当空间频率。因此，更窄的条带对应更宽的衍射图样，反映了傅里叶分析中固有的互逆展宽——这一概念类似于海森堡不确定性原理。

10. Comparison with Double-Slit and Grating Patterns | 与双缝和光栅图样的比较

The uniform strip produces a pure diffraction pattern, whereas a double-slit setup yields an interference pattern modulated by a diffraction envelope. Key differences are summarised below:

均匀条带产生纯粹的衍射图样，而双缝设置产生受衍射包络调制的干涉图样。主要区别总结如下：

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