📚 Fourier Transform in Radiation Problems | 辐射问题中的傅里叶变换
Radiation phenomena—from thermal heat waves to electromagnetic signals—are often described by functions of time or space. The Fourier transform is a mathematical bridge that converts such functions into their frequency components, revealing the spectral content of radiation. In IB Mathematics, this integral transform appears in extended topics such as Fourier analysis and differential equations. This article explains how the Fourier transform is applied to radiation problems, with clear definitions, properties, and worked examples that align with the IB syllabus.
辐射现象——从热波到电磁信号——通常用时间或空间的函数来描述。傅里叶变换是一座数学桥梁，它将这类函数转换成频率分量，揭示辐射的频谱内容。在 IB 数学中，这一积分变换出现在傅里叶分析和微分方程等拓展专题中。本文以清晰的定义、性质和与 IB 考点一致的例题，讲解傅里叶变换如何应用于辐射问题。
1. From Time to Frequency: The Core Idea | 从时间到频率：核心思想
A radiation source often produces a signal \( f(t) \) that varies in time. For example, a hot object emits electromagnetic waves whose electric field oscillates rapidly. Instead of analysing the signal directly, we can ask: which frequencies are present, and with what amplitudes? The Fourier transform answers this by decomposing \( f(t) \) into a continuous spectrum of sinusoidal waves \( e^{i\omega t} \).
辐射源通常产生随时间变化的信号 \( f(t) \)。例如，一个发热物体发射电磁波，其电场迅速振荡。与其直接分析信号，我们可以问：信号中包含哪些频率？各自的振幅是多少？傅里叶变换通过将 \( f(t) \) 分解为连续频谱的正弦波 \( e^{i\omega t} \) 来回答这一问题。
In radiation physics, this decomposition is not just a mathematical trick. It corresponds to the way a prism splits white light into colours, or how a radio receiver separates different stations. The Fourier transform is the mathematical prism.
在辐射物理中，这种分解不仅仅是数学技巧。它对应着棱镜将白光分成不同颜色的过程，也对应着无线电接收器分离不同电台的方式。傅里叶变换就是数学意义上的棱镜。
2. Defining the Fourier Transform | 傅里叶变换的定义
For an integrable function \( f(t) \), the Fourier transform \( F(\omega) \) is defined as an integral over all time:
对于一个可积函数 \( f(t) \)，傅里叶变换 \( F(\omega) \) 定义为在整个时间域上的积分：
F(ω) = ∫-∞∞ f(t) e-iωt dt
Here \( \omega \) is the angular frequency (radians per second), and \( i \) is the imaginary unit. The factor \( e^{-i\omega t} \) acts as a “probe” that checks how much of frequency \( \omega \) is contained in \( f(t) \).
其中 \( \omega \) 是角频率（弧度/秒），\( i \) 是虚数单位。因子 \( e^{-i\omega t} \) 就像一个“探针”，检测 \( f(t) \) 中包含多少频率 \( \omega \) 的分量。
In IB notation, we often use \( x \) instead of \( t \), and \( k \) instead of \( \omega \). The definition remains valid for spatial variables, which is useful for radiation patterns along a line or a surface.
在 IB 记法中，我们常用 \( x \) 代替 \( t \)，用 \( k \) 代替 \( \omega \)。该定义对空间变量同样成立，这对于分析沿直线或表面的辐射分布非常有用。
3. The Inverse Transform | 逆变换
The Fourier transform is invertible. Given the frequency spectrum \( F(\omega) \), the original signal can be recovered by the inverse formula:
傅里叶变换是可逆的。给定频谱 \( F(\omega) \)，原始信号可以通过逆变换公式恢复：
f(t) = (1/2π) ∫-∞∞ F(ω) eiωt dω
The factor \( 1/2\pi \) normalises the integral, accounting for the total contribution of all frequencies. In some conventions the factor is split symmetrically between the transform and its inverse. Always check the convention used by your exam board.
因子 \( 1/2\pi \) 用于归一化积分，表示所有频率的总贡献。在某些约定中，该因子会对称地分配到正变换和逆变换中。请务必留意你所在考试局采用的约定。
This inverse property is essential in radiation problems: a measured spectrum can be transformed back into a time-domain waveform, for example reconstructing a radiation pulse from its observed frequency components.
逆变换性质在辐射问题中至关重要：测得的频谱可以变换回时域波形，例如从观测到的频率分量重建一个辐射脉冲。
4. Key Properties of the Transform | 变换的关键性质
The Fourier transform has several properties that simplify calculations. Let \( \mathcal{F}\{f(t)\} = F(\omega) \) and \( \mathcal{F}\{g(t)\} = G(\omega) \).
傅里叶变换具有多个简化计算的特性。设 \( \mathcal{F}\{f(t)\} = F(\omega) \)，\( \mathcal{F}\{g(t)\} = G(\omega) \)。
- Linearity: \( \mathcal{F}\{af(t) + bg(t)\} = aF(\omega) + bG(\omega) \).
- Time shift: \( \mathcal{F}\{f(t – t_0)\} = e^{-i\omega t_0} F(\omega) \).
- Frequency shift: \( \mathcal{F}\{e^{i\omega_0 t} f(t)\} = F(\omega – \omega_0) \).
- Scaling: \( \mathcal{F}\{f(at)\} = (1/|a|) F(\omega/a) \).
- Derivative: \( \mathcal{F}\{f'(t)\} = i\omega F(\omega) \).
- 线性性质： \( \mathcal{F}\{af(t) + bg(t)\} = aF(\omega) + bG(\omega) \)。
- 时移性质： \( \mathcal{F}\{f(t – t_0)\} = e^{-i\omega t_0} F(\omega) \)。
- 频移性质： \( \mathcal{F}\{e^{i\omega_0 t} f(t)\} = F(\omega – \omega_0) \)。
- 缩放性质： \( \mathcal{F}\{f(at)\} = (1/|a|) F(\omega/a) \)。
- 导数性质： \( \mathcal{F}\{f'(t)\} = i\omega F(\omega) \)。
The derivative property is especially powerful. It converts differentiation into multiplication by \( i\omega \), turning differential equations into algebraic equations.
导数性质尤其强大。它将微分运算转换为乘以 \( i\omega \)，从而把微分方程转化为代数方程。
5. Convolution: When Radiation Interacts | 卷积：当辐射相互作用时
When radiation passes through a medium, the observed signal is often a convolution of the original radiation with the medium’s response function \( h(t) \):
当辐射穿过介质时，观测到的信号通常是原始辐射与介质响应函数 \( h(t) \) 的卷积：
(f * h)(t) = ∫-∞∞ f(τ) h(t – τ) dτ
The convolution theorem states that the Fourier transform of a convolution is the product of the individual transforms:
卷积定理指出：卷积的傅里叶变换等于各自变换的乘积：
ℱ{f * h} = F(ω) H(ω)
This is why many radiation measuring instruments first obtain the product in the frequency domain and then apply an inverse Fourier transform to recover the deconvolved signal.
这就是为什么许多辐射测量仪器首先在频域中得到乘积，然后再用逆傅里叶变换恢复解卷积后的信号。
In spectroscopy, this principle is used to correct instrument broadening: the measured spectrum is divided by the instrument response in frequency space.
在光谱学中，该原理用于校正仪器展宽：在频率空间中将实测频谱除以仪器响应函数。
6. Solving Differential Equations via Fourier Transform | 用傅里叶变换求解微分方程
Radiation problems often lead to linear differential equations. Consider the one-dimensional heat equation, which models thermal radiation diffusion:
辐射问题通常归结为线性微分方程。考虑一维热方程，它用于模拟热辐射扩散：
∂u/∂t = α ∂²u/∂x²
Applying the Fourier transform with respect to \( x \), we let \( U(k,t) = \mathcal{F}\{u(x,t)\} \). The spatial derivative becomes \( -k^2 U(k,t) \), so the equation reduces to an ordinary differential equation in \( t \):
对 \( x \) 进行傅里叶变换，设 \( U(k,t) = \mathcal{F}\{u(x,t)\} \)。空间导数变为 \( -k^2 U(k,t) \)，因此方程简化为关于 \( t \) 的常微分方程：
dU/dt = -α k² U
Its solution is \( U(k,t) = U(k,0) e^{-\alpha k^2 t} \). Finally, applying the inverse transform gives the temperature distribution at any time.
其解为 \( U(k,t) = U(k,0) e^{-\alpha k^2 t} \)。最后，应用逆变换即可得到任意时刻的温度分布。
This method works because the Fourier transform replaces differentiation by multiplication, significantly simplifying the problem.
这种方法之所以有效，是因为傅里叶变换将微分运算替换为乘法运算，从而大大简化了问题。
7. The Heat Equation and Thermal Radiation | 热方程与热辐射
In thermal radiation, the heat equation describes how temperature changes propagate through a material. A classic initial condition is a point source at \( x = 0 \), represented by a Dirac delta function \( u(x,0) = Qδ(x) \).
在热辐射中，热方程描述温度变化如何在材料中传播。一个经典的初始条件是位于 \( x = 0 \) 的点热源，表示为狄拉克 delta 函数 \( u(x,0) = Qδ(x) \)。
The Fourier transform of a delta function is constant: \( U(k,0) = Q \). Substituting into the solution above yields:
delta 函数的傅里叶变换是常数：\( U(k,0) = Q \)。代入上述解可得：
U(k,t) = Q e-αk²t
The inverse transform of a Gaussian is another Gaussian, giving the familiar heat kernel:
高斯函数的逆变换仍是高斯函数，于是得到熟知的的热核：
u(x,t) = Q / (2√(παt)) · e-x²/(4αt)
This formula shows how heat spreads: the amplitude decreases as \( 1/√t \), while the width of the distribution grows as \( √t \).
该公式展示了热量如何扩散：振幅随 \( 1/√t \) 衰减，而分布的宽度随 \( √t \) 增大。
In IB problems, you may be asked to verify this solution by differentiation or to sketch the temperature profile at different times.
在 IB 题目中，你可能会被要求通过求导验证该解，或绘制不同时刻的温度分布图。
8. Blackbody Radiation Spectrum | 黑体辐射谱
A blackbody emits radiation whose spectral energy density depends on frequency and temperature. The Planck law gives:
黑体辐射的频谱能量密度取决于频率和温度。普朗克定律给出：
B(ν,T) = (2hν³/c²) · 1/(ehν/kT – 1)
Although this formula can be derived using quantum statistics, the Fourier transform plays a practical role in measuring such spectra. In a Fourier-transform infrared (FTIR) spectrometer, the instrument records an interferogram \( I(x) \) as a function of mirror displacement \( x \).
虽然该公式可以通过量子统计推导，但傅里叶变换在测量这类光谱时起着实际作用。在傅里叶变换红外（FTIR）光谱仪中，仪器记录的是随反射镜位移 \( x \) 变化的干涉图 \( I(x) \)。
The spectrum \( S(ν) \) is obtained by taking the Fourier transform of the interferogram:
光谱 \( S(ν) \) 通过对干涉图进行傅里叶变换得到：
S(ν) = ∫-∞∞ I(x) e-i2πνx dx
This demonstrates a direct application of the transform to radiation analysis. The example also shows why understanding the inverse transform is important for reconstructing actual spectra from measured data.
这展示了傅里叶变换在辐射分析中的直接应用。该例子也说明了为什么理解逆变换对于从测量数据重建真实光谱非常重要。
9. Electromagnetic Wave Propagation | 电磁波传播
Electromagnetic radiation in free space is governed by the wave equation:
自由空间中的电磁辐射由波动方程控制：
∂²E/∂x² = (1/c²) ∂²E/∂t²
Taking the Fourier transform with respect to \( x \) and \( t \), the equation becomes \( k²E = (ω²/c²)E \), leading to the dispersion relation \( ω = ±ck \).
对 \( x \) 和 \( t \) 进行傅里叶变换，方程变为 \( k²E = (ω²/c²)E \)，从而得到色散关系 \( ω = ±ck \)。
For a non-monochromatic pulse, the Fourier transform expresses the pulse as a superposition of plane waves of different frequencies. Each component travels at its own phase velocity; in a dispersive medium the pulse shape changes as it propagates.
对于非单色脉冲，傅里叶变换将脉冲表示为不同频率平面波的叠加。每个分量以各自的相速度传播；在色散介质中，脉冲形状会在传播过程中发生变化。
IB extended essay projects or option topics often use Fourier methods to derive the group velocity \( v_g = dω/dk \), a key concept for radiation in media.
IB 拓展论文或选修专题中，常使用傅里叶方法推导群速度 \( v_g = dω/dk \)，这是介质中辐射传播的关键概念。
10. Gaussian Pulses and Packet Spreading | 高斯脉冲与波包扩展
A common model for a radiation pulse is a Gaussian envelope:
辐射脉冲的常见模型是高斯包络：
f(t) = A e-t²/(2σ²)
Its Fourier transform is also a Gaussian:
其傅里叶变换也是高斯函数：
F(ω) = A√(2πσ²) e-σ²ω²/2
This leads to the important relation between pulse duration and spectral width: \( \Delta \omega \Delta t \ge 1/2 \).
由此得出脉冲持续时间与频谱宽度之间的重要关系：\( \Delta \omega \Delta t \ge 1/2 \)。
In radiation physics, this means that a very short pulse necessarily has a broad frequency spectrum. This is why femtosecond laser pulses contain many colours, even if the gain medium is narrowband.
在辐射物理中，这意味着极短的脉冲必然具有较宽的频谱。这正是飞秒激光脉冲包含多种颜色的原因，即使增益介质本身是窄带的。
The product \( \Delta \omega \Delta t \) is called the time-bandwidth product. It has a minimum value for Gaussian pulses, which are therefore “transform-limited”.
乘积 \( \Delta \omega \Delta t \) 称为时间带宽积。高斯脉冲具有最小的时间带宽积，因此被称为“变换极限”脉冲。
11. Sampling and Discrete Fourier Transform in Radiation Detection | 辐射探测中的采样与离散傅里叶变换
In practice, radiation signals are sampled at discrete times. The discrete Fourier transform (DFT) approximates the continuous transform:
在实际中，辐射信号是按照离散时间采样的。离散傅里叶变换（DFT）近似连续变换：
Xk = Σn=0N-1 xn e-i2πkn/N
The fast Fourier transform (FFT) algorithm computes this efficiently, enabling real-time radiation spectrum analysis.
快速傅里叶变换（FFT）算法能高效计算该和式，从而支持实时辐射频谱分析。
The Nyquist sampling theorem states that a signal must be sampled at at least twice its highest frequency component to avoid aliasing:
奈奎斯特采样定理指出，为避免混叠，采样率必须至少是信号最高频率分量的两倍：
fs ≥ 2 fmax
Violating this condition causes high-frequency radiation components to appear as lower frequencies, corrupting the spectral measurement.
违反该条件会导致高频辐射分量被折叠为低频，从而破坏频谱测量。
IB students should be familiar with this application, as it connects discrete mathematics with physical measurement.
IB 学生应熟悉这一应用，因为它将离散数学与物理测量联系起来。
12. Conclusion | 总结
The Fourier transform is an indispensable tool in the study of radiation. It allows us to move between time/space domain and frequency domain, simplifies differential equations, and explains fundamental phenomena such as pulse broadening and spectral measurement.
傅里叶变换是研究辐射不可或缺的工具。它使我们能够在时间/空间域与频率域之间转换，简化微分方程，并解释脉冲展宽和光谱测量等基本现象。
For IB Mathematics students, mastering the definition, inverse formula, and key properties of the Fourier transform opens the door to solving realistic problems in physics and engineering. Practice by transforming simple functions such as the Gaussian, the rectangular pulse, and the delta function, and verify the convolution theorem explicitly.
对于 IB 数学学生而言，掌握傅里叶变换的定义、逆变换公式和关键性质，为解决物理和工程中的实际问题打开了大门。练习时，可以对高斯函数、矩形脉冲和 delta 函数等简单函数进行变换，并明确验证卷积定理。
With this foundation, you can confidently approach advanced radiation problems and appreciate the deep unity between mathematics and the physical world.
有了这个基础，你就能自信地处理高级辐射问题，并体会到数学与物理世界之间深刻的统一性。
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