Experiment 5: Polarization and Interference # 实验5：偏振和干涉

Nate Saffold <br>nas2173@columbia.edu

Office Hour: Mondays, 5:30PM-6:30PM @ Pupin 1216

奈特·萨福尔德 <br>nas2173@columbia.edu

办公时间：周一，5:30PM-6:30PM @ 普平楼 1216

INTRO TO EXPERIMENTAL PHYS-LAB 1493/1494/2699 # 实验物理实验室导论 1493/1494/2699

Introduction # 介绍

• Outline:
• Review of physics:

1. Electromagnetic waves
2. Polarization of an EM wave
3. Interference and diffraction (double and single slit experiments)

• Polarization and Interference experiment:

1. Description of the apparati
2. Data analysis

• Tips for the experiment
• 大纲：
• 物理学的回顾：

1. 电磁波
2. 电磁波的偏振
3. 干涉和衍射（双缝和单缝实验）

• 偏振和干涉实验：

1. 仪器的描述
2. 数据分析

• 实验的提示

Electromagnetic waves # 电磁波

• Electromagnetic wave = oscillating electric and magnetic fields
• An EM wave propagates in vacuum at the speed of light

$$c=299792458 \mathrm{~m} / \mathrm{s} \simeq 3 \times 10^{8} \mathrm{~m} / \mathrm{s}$$

• Electric and magnetic fields are always perpendicular to each other
• 电磁波 = 振荡的电场和磁场
• 电磁波在真空中以光速传播

$$c=299792458 \mathrm{~m} / \mathrm{s} \simeq 3 \times 10^{8} \mathrm{~m} / \mathrm{s}$$

• 电场和磁场总是相互垂直的

(James Clerk Maxwell... Pretty smart guy...)

(詹姆斯·克拉克·麦克斯韦...非常聪明的人...)

Polarization # 偏振

• Polarization of a light wave = direction of oscillation of the electric field
• Every day light is usually unpolarized. All directions of the electric field are equally probable.
• Linearly polarized light = the direction of oscillation at a particular point in space is always the same
• 光波的偏振 = 电场的振动方向
• 日常的光通常是非偏振的。电场的所有方向都是等概率的。
• 线偏振光 = 在特定空间的点上，振动的方向始终相同

Unpolarized beam of light moving perpendicularly to the screen. No preferred direction of oscillation

Linearly polarized beam of light moving perpendicularly to the screen. Electric field points in one direction only.

非偏振光束垂直于屏幕移动。没有优先的振动方向

线偏振光束垂直于屏幕移动。电场仅指向一个方向。

Polarization by selective absorption # 通过选择性吸收的偏振

• White light (as from a light bulb) is usually unpolarized.
• Polarizer = material that selects one particular direction of oscillation of the incoming light
• If we place two linear polarizers in sequence then:

1. First one: it makes unpolarized light linearly polarized
2. Second one: it determines which fraction of the incoming light arrives at the end of the apparatus

• 白光（如来自灯泡的光）通常是非偏振的。
• 偏振片 = 选择入射光的特定振动方向的材料
• 如果我们按顺序放置两个线性偏振片，那么：

1. 第一个：它使非偏振光变成线偏振光
2. 第二个：它决定入射光的哪个部分到达装置的末端

Polarized and different intensity

偏振且强度不同

Malus' Law # 马吕斯定律

• If we place two polarizers in sequence, the magnitude of the transmitted polarized electric field will depend on the angle between polarizers.
• 如果我们按顺序放置两个偏振片，透射的偏振电场的大小将取决于偏振片之间的角度。

Electric field coming out of first polarizer

Orientation of second polarizer

Orientation of first polarizer

• This is how the electric field behaves when passing through two linear polarizers. What do we actually see?

从第一个偏振片出来的电场

第二个偏振片的方向

第一个偏振片的方向

• 这就是电场通过两个线性偏振片时的行为。我们实际看到的是什么？

Malus' Law # 马吕斯定律

• Our eyes cannot detect the electric field directly
• We only see its time average $\longrightarrow$ intensity
• Intensity = magnitude squared of the electric field

$$I=|\vec{E}|^{2}$$

• From the formula in the previous slide we obtain the socalled Malus' Law:
• 我们的眼睛不能直接检测电场
• 我们只能看到它的时间平均值 $\longrightarrow$ 强度
• 强度 = 电场的大小的平方

$$I=|\vec{E}|^{2}$$

• 从前一个幻灯片中的公式，我们得到所谓的马吕斯定律：

Intensity coming out of second polarizer

从第二个偏振片出来的强度

Malus' Law # 马吕斯定律

• The intensity coming out of the polarizers as a function of the angle between the two then looks like:
• 从偏振片出来的强度作为两者之间的角度的函数看起来像：

Interference # 干涉

• An essential property of waves is the ability to get combined with other waves
• The result of this superposition can lead to a wave with greater or smaller amplitude
• This phenomenon is called interference
• 波的一个基本特性是与其他波结合的能力
• 这种叠加的结果可能导致具有更大或更小振幅的波
• 这种现象被称为干涉

Young's double slit experiment # 杨氏双缝实验

• Question: if light is a wave, what should we expect from it?
• Young's double-slit experiment (1801):
• Pass light through two very narrow slits and observe pattern on distant screen.
• Waves coming out of the two slits interfere and create a fringe pattern of bright and dark spots
• 问题：如果光是一种波，我们应该期待什么？
• 杨氏双缝实验（1801年）：
• 让光通过两个非常窄的缝隙，并在远处的屏幕上观察图案。
• 从两个缝隙出来的波相互干涉，创造出明暗斑点的条纹图案

Young's double slit experiment # 杨氏双缝实验

• The pattern on the wall can be derived from optics
• We need to see how two rays from the two slits interfere with each other
• 墙上的图案可以从光学推导出来
• 我们需要看看来自两个缝隙的两条光线如何相互干涉

Condition for a bright spot # 明亮斑点的条件

• Let the screen be at a distance $D$ from the slits. Let's take this distance much larger than the distance between slits ( $D \gg d$ ).
• Rays emerge almost parallel to each other.
• In order for both of them to end up on the same point (and hence interfere) one of the two must travel a slightly longer distance:
• 让屏幕位于距离缝隙的距离 $D$ 处。让我们假设这个距离远大于缝隙之间的距离（$D \gg d$）。
• 光线几乎平行地射出。
• 为了使两者都到达同一个点（从而产生干涉），其中一个必须行进稍长的距离：

$$\Delta \ell=d \sin \theta$$

Condition for a bright spot # 明亮斑点的条件

• We want a bright spot, i.e. constructive interference
• In order for two maxima to overlap the difference in travel distance must be a multiple of the wavelength
• 我们想要一个明亮的斑点，即建设性干涉
• 为了使两个极大值重叠，行进距离的差异必须是波长的倍数

$$\Delta \ell=d \sin \theta=m \lambda$$

where $m$ is an integer number and $\lambda$ is the wavelength of incoming light

其中 $m$ 是一个整数，$\lambda$ 是入射光的波长

• Since $D \gg d$ we can use the small angle approximation:
• 由于 $D \gg d$，我们可以使用小角度近似：

$\sin \theta \simeq \tan \theta \simeq x_{m} / D$

The distance of the $m$-th bright spot from the center is then:

$m$ 阶明亮斑点距离中心的距离是：

$x_{m}=m\left(\frac{\lambda D}{d}\right)$

Young's double-slit experiment # 杨氏双缝实验

• The pattern appearing in the double slit experiment is due to the interference effect
• However, in real life you will also have diffraction. Light phenomena will always be a combination of the two effects
• 双缝实验中出现的图案是由于干涉效应
• 然而，在实际生活中，你还会有衍射。光现象总是两种效应的组合

Young's double-slit experiment # 杨氏双缝实验

• The pattern appearing in the double slit experiment is due to the interference effect
• However, in real life you will also have diffraction. Light phenomena will always be a combination of the two effects
• Diffraction = Spread of waves around an obstacle or slit
• Only relevant when the obstacle size is comparable to the wavelength
• Wave behaves as if it was interfering with itself
• 双缝实验中出现的图案是由于干涉效应
• 然而，在实际生活中，你还会有衍射。光现象总是两种效应的组合
• 衍射 = 波在障碍物或缝隙周围的扩散
• 只有当障碍物尺寸与波长相当时才相关
• 波的行为就像是与自身干涉

Diffraction intensity pattern # 衍射强度图案

• If diffraction is made around a single slit the intensity is given by:
• 如果在单个缝隙周围产生衍射，则强度由以下公式给出：

$$\left.I=I_{0}\left(\frac{\sin \left(\frac{\pi a}{\lambda} \sin \theta\right)}{\frac{\pi a}{\lambda} \sin \theta}\right)\right)^{2=\text { single-slit width }} \begin{aligned} & \lambda=\text { wavelength } \\ & \theta=\text { angle on the screen } \end{aligned}$$

Diffraction intensity pattern # 衍射强度图案

• If diffraction is made around a single slit the intensity is given by:
• 如果在单个缝隙周围产生衍射，则强度由以下公式给出：

$$I=I_{0}\left(\frac{\sin \left(\frac{\pi a}{\lambda} \sin \theta\right)}{\frac{\pi a}{\lambda} \sin \theta}\right)$$

a = single-slit width $\lambda=$ wavelength $\theta=$ angle on the screen

a = 单缝宽度 $\lambda=$ 波长 $\theta=$ 屏幕上的角度

• Single-slit minima occur at:
• 单缝极小值出现在：

$$\begin{aligned} & \frac{\pi a}{\lambda} \sin \theta=n \pi \\ & n= \pm 1, \pm 2, \pm 3, \ldots \end{aligned}$$

• Again, using small angle approximation, the minima occur at:
• 再次，使用小角度近似，极小值出现在：

$$x_{n}=n\left(\frac{\lambda D}{a}\right)$$

Diffiraction intensity pattern # 衍射强度图案

• If diffraction is made around a single slit the intensity is given by:
• 如果在单个缝隙周围产生衍射，则强度由以下公式给出：

$$I=I_{0}\left(\frac{\sin \left(\frac{\pi a}{\lambda} \sin \theta\right)}{\frac{\pi a}{\lambda} \sin \theta}\right)^{2}$$

a = single-slit width $\lambda=$ wavelength $\theta=$ angle on the screen

a = 单缝宽度 $\lambda=$ 波长 $\theta=$ 屏幕上的角度

• Single-slit minima occur at:
• 单缝极小值出现在：

$$\begin{aligned} & \frac{\pi a}{\lambda} \sin \theta=n \pi \\ & n= \pm 1, \pm 2, \pm 3, \ldots \end{aligned}$$

• Again, using small angle approximation, the minima occur at:
• 再次，使用小角度近似，极小值出现在：

The Experiment # 实验

Goals # 目标

• To study the polarization properties of EM waves
• Polarize a beam of white light
• Verify Malus' Law: measure intensity of white light when light travels through two polarizers at an angle $\theta$ with respect to each other:
• 研究电磁波的偏振特性
• 使白光的光束偏振
• 验证马吕斯定律：测量当光通过两个相对成角度 $\theta$ 的偏振片时白光的强度：

$$I(\theta)=I_{0} \cos ^{2} \theta$$

• To study the interference and diffraction phenomena
• Use a He-Ne laser as coherent light source (single wavelength)
• Determine $\lambda$ of the laser from the double-slit interference pattern
• Determine $\lambda$ of the laser from single-slit diffraction pattern
• 研究干涉和衍射现象
• 使用氦氖激光器作为相干光源（单一波长）
• 从双缝干涉图案确定激光器的 $\lambda$
• 从单缝衍射图案确定激光器的 $\lambda$

Equipment # 设备

Part 1: Malus' law # 第1部分：马吕斯定律

• Equipment:
• Incandescent light source (source of EM waves)
• Polarization filters
• Photometer (measures intensity of outgoing EM waves)
• For polarization:
• Incandescent light is a source of unpolarized EM waves - E-field has no preferred direction
• Place a polarizer in front of the incandescent light source
• 设备：
• 白炽光源（电磁波的来源）
• 偏振滤光片
• 光度计（测量输出电磁波的强度）
• 关于偏振：
• 白炽光是非偏振电磁波的来源 - 电场没有优先的方向
• 在白炽光源前放置一个偏振片

Outgoing light should be polarized! # 输出的光应该是偏振的！

Part 1: Malus' Law # 第1部分：马吕斯定律

• Polarizers in sequence
• Place two polarizers in front of the light source
• Measuring intensity of light:
• Align the axis of both polarizers such that they are parallel to each other. Measure and record intensity.
• Rotate the second polarizer in 5-10 degree increments with respect to the first one. Record intensity at each step
• 偏振片按顺序排列
• 在光源前放置两个偏振片
• 测量光的强度：
• 对齐两个偏振片的轴，使它们彼此平行。测量并记录强度。
• 相对于第一个，以5-10度的增量旋转第二个偏振片。在每个步骤记录强度

Part 1: Malus' Law # 第1部分：马吕斯定律

• According to Malus' law:
• 根据马吕斯定律：

$$I(\theta)=I_{0} \cos ^{2} \theta$$

• You will have a set of $\left(I_{i}, \theta_{i}\right)$ pairs
• Linearize the data by plotting $I / I_{0}$ vs. $\cos ^{2} \theta$
• Perform a linear fit and find slope and intercept (w/ errors!). Are they what you expect?
• Plot residuals to check for consistency of the fit
• 你将有一组$\left(I_{i}, \theta_{i}\right)$数据对
• 通过绘制$I / I_{0}$与$\cos ^{2} \theta$的关系图来线性化这些数据
• 执行线性拟合并找出斜率和截距（带误差！）。它们是否符合你的预期？
• 绘制残差以检查拟合的一致性

Part 2: double-slit experiment # 第2部分：双缝实验

• Procedure:
• Mount laser in far end of optical bench.
• Mount slit set C (double-slit) in front of the laser beam.
• Observe double-slit intensity pattern with a white piece of paper.
• Use linear translator with fiber optic attached to measure intensity at different positions in the transverse direction
• 步骤：
• 将激光器安装在光学平台的远端。
• 在激光束前安装缝隙组C（双缝）。
• 用一张白纸观察双缝强度图案。
• 使用连接了光纤的线性平移器在横向的不同位置测量强度

Part 2: double-slit experiment # 第2部分：双缝实验

• Record the position of maxima
• Plot $x_{m}$ vs. order number ( $m$ )
• Perform a linear fit
• Use slope to estimate the wavelength of the laser
• Remember to propagate uncertainties!
• If you are careful enough the results can be fairly accurate:
• 记录极大值的位置
• 绘制$x_{m}$与级数（$m$）的关系图
• 执行线性拟合
• 使用斜率来估计激光器的波长
• 记得传播不确定性！
• 如果你足够仔细，结果可以相当准确：

$$\begin{aligned} & \lambda_{\text {meas }}=626.1 \pm 3.5 \mathrm{~nm} \\ & \lambda_{\text {theor }}=632.8 \mathrm{~nm} \end{aligned}$$

Part 3: single-slit envelope # 第3部分：单缝包络

• Once again take measurements in the transverse direction but in smaller increments and look for brighter spots
• Plot relative intensity $\left(I / I_{0}\right)$ vs. order number $m$.
• Should be able to observe the single-slit envelope.
• Using the single-slit width (a), determine wavelength of laser.
• 再次在横向进行测量，但使用更小的增量，寻找更亮的斑点
• 绘制相对强度$\left(I / I_{0}\right)$与级数$m$的关系图。
• 应该能够观察到单缝包络。
• 使用单缝宽度（a），确定激光器的波长。

$$I=I_{0}\left(\frac{\sin \left(\frac{\pi a}{\lambda} \sin \theta\right)}{\frac{\pi a}{\lambda} \sin \theta}\right)^{2}$$

IMPORTANT: For this part of the experiment switch to slit D!

重要：对于这部分实验，切换到缝隙D！

Part 3: single-slit envelope # 第3部分：单缝包络

• Once again take measurements in the transverse direction but in smaller increments and look for brighter spots
• Plot relative intensity $\left(I / I_{0}\right)$ vs. position $x_{m}$.
• Should be able to observe the single-slit envelope, note where the single slit minima are $x_{n}$
• Plot $x_{n}$ vs. $n$, where $n$ is minima order number
• Using slope and wavelength from part 2, determine a
• 再次在横向进行测量，但使用更小的增量，寻找更亮的斑点
• 绘制相对强度$\left(I / I_{0}\right)$与位置$x_{m}$的关系图。
• 应该能够观察到单缝包络，注意单缝极小值$x_{n}$的位置
• 绘制$x_{n}$与$n$的关系图，其中$n$是极小值级数
• 使用第2部分的斜率和波长，确定a

NOTE: This part may vary depending on your TA. Ask them for the exact procedure!

注意：这部分可能会根据你的助教而有所不同。向他们询问确切的步骤！

Tips # 提示

• Here are some tips that might be useful:
• 以下是一些可能有用的提示：

1. For all the three parts: be careful when you read the intensity. The photometer is analogical and the reading might be influenced by parallax. Try to read the photometer always in the same way
2. For the polarizer part: put the polarizers as close as possible to the optic fiber cable to minimize the amount of environmental light coming in.
3. For the last part: move everything closer to the laser to make the pattern bigger but remember to change the value of $D$ !
4. 对于所有三个部分：在读取强度时要小心。光度计是模拟的，读数可能会受到视差的影响。尽量始终以相同的方式读取光度计
5. 对于偏振片部分：将偏振片尽可能靠近光纤电缆，以最小化进入的环境光量。
6. 对于最后一个部分：将所有东西移近激光器以使图案变大，但记得更改$D$的值！