Experiment 6 Polarization and Interference 实验 6 偏振和干涉

1 Purpose 1 目的

The purpose of this experiment is to investigate two basic concepts of wave mechanics: polarization and interference. You will conduct the experiment with light, but the general concepts apply to many other types of waves. 本实验的目的是研究波动力学的两个基本概念：偏振和干涉。您将使用光进行实验，但这些一般概念适用于许多其他类型的波。

2 Introduction 2 介绍

2.1 Polarization 2.1 偏振

Within the enormous range of wave phenomena observed in nature, we can identify two basic types of waves: 在自然中观察到的庞大波动现象范围内，我们可以识别两种基本类型的波：

1. Longitudinal: waves whose vibrations are parallel to the direction of wave propagation. When a longitudinal wave moves through a medium, one observes density variations - regions of compression and rarefaction - in the material. Sound is an example of a longitudinal wave.
2. 纵波：振动方向与波传播方向平行的波。当纵波通过介质时，人们会观察到材料中的密度变化 - 压缩和稀疏的区域。声音是纵波的一个例子。
3. Transverse: waves whose vibrations (or displacements) are perpendicular to the direction of propagation. Light is an example of a transverse wave.
4. 横波：振动（或位移）垂直于传播方向的波。光是横波的一个例子。

Polarization occurs only in transverse waves. It refers to the axis of the waves' oscillation, which is always perpendicular to the wave direction. For a wave traveling in three spatial dimensions, the direction of polarization can occur anywhere within the plane perpendicular to the direction of propagation. If a wave oscillates along only one axis within this plane, it is said to be polarized. 偏振只发生在横波中。它指的是波的振动轴，该轴始终垂直于波的方向。对于在三个空间维度中传播的波，偏振的方向可以发生在垂直于传播方向的平面内的任何位置。如果波仅沿着这个平面内的一个轴振动，则称其为偏振的。

Light is electromagnetic radiation; it consists of oscillating electric and magnetic fields that vibrate perpendicular to the direction of the wave. When we speak of the polarization state of light, we refer specifically to the vibration axis of the electric field. 光是电磁辐射；它由垂直于波的方向振动的电场和磁场组成。当我们谈论光的偏振状态时，我们特指电场的振动轴。

Typically, light produced by an incandescent bulb or a candle is unpolarized; that is, the electric field associated with the emitted light oscillates in random directions within the plane perpendicular to the direction of propagation. However, unpolarized light can become polarized under several circumstances. One of these occurs when we pass unpolarized light through a polarizing filter, as shown in Fig. 6.1. For visible light, such filters are typically made of polymer films, plastics whose molecules form long chains oriented along one axis. 通常，由白炽灯泡或蜡烛产生的光是非偏振的；即与发射光相关的电场在垂直于传播方向的平面内沿随机方向振动。然而，非偏振光可以在几种情况下变成偏振光。其中一种情况是当我们将非偏振光通过偏振滤光片时，如图6.1所示。对于可见光，这类滤光片通常由聚合物薄膜制成，这些塑料的分子沿一个轴形成长链。

When an unpolarized light wave encounters the filter, the component of the electric field oscillating parallel to the molecular chains is absorbed in the material. The component of the field perpendicular to the molecules' long axes is transmitted. Therefore, the direction perpendicular to the molecular chains is called the transmission axis. 当非偏振光波遇到滤光片时，平行于分子链振动的电场的分量被材料吸收。垂直于分子长轴的场的分量被传输。因此，垂直于分子链的方向被称为透射轴。

Figure 6.1: Creation and detection of polarized light. 图6.1：偏振光的产生和检测。

Suppose a wave has been polarized by a filter such that its polarization is characterized by an electric field $\vec{E}_{0}$. We can analyze the polarization state by passing the light through a second filter (called an analyzer) whose transmission axis is oriented at an angle $\theta$ with respect to the first filter. The effect of the analyzer is to pick off the component of $\vec{E}_{0}$ parallel to its transmission axis, $E_{0} \cos \theta$. The remaining light is absorbed. In this experiment, we use a laser, which is a polarized light source, and shine the light through a polarizer (the analyzer), recreating the physics of two successive polarizers. 假设一个波已经被滤光片偏振，使其偏振由电场 $\vec{E}_{0}$ 表征。我们可以通过将光通过第二个滤光片（称为分析器）来分析偏振状态，该分析器的透射轴相对于第一个滤光片以角度 $\theta$ 定向。分析器的效果是选取平行于其透射轴的 $\vec{E}_{0}$ 的分量，即 $E_{0} \cos \theta$。剩余的光被吸收。在本实验中，我们使用激光器，它是一种偏振光源，并将光通过偏振器（分析器），重现两个连续偏振器的物理过程。

Since the intensity of the light wave is proportional to $|\vec{E}|^{2}$, the intensity of light after it has passed through the analyzer is 由于光波的强度与 $|\vec{E}|^{2}$ 成正比，光通过分析器后的强度为

$$\begin{equation*} I=I_{0} \cos ^{2} \theta, \tag{6.1} \end{equation*}$$

where $I_{0}=\left|\vec{E}_{0}\right|^{2}$. This shift in the intensity, known as Malus' Law, can be detected by a human eye or a photometer. 其中 $I_{0}=\left|\vec{E}_{0}\right|^{2}$。这种强度的变化，被称为马吕斯定律，可以通过人眼或光度计检测到。

2.2 Interference: Young's Double Slit 2.2 干涉：杨氏双缝

Another key property of waves is their ability to superimpose. That is, when two waves encounter each other in a medium, the resulting wave is simply the algebraic sum of the two individual waves. The combination of two or more waves into a third is called interference. Interference can occur constructively or destructively: 波的另一个关键特性是它们的叠加能力。也就是说，当两个波在介质中相遇时，产生的波仅仅是两个单独波的代数和。两个或更多波合成第三个波的组合被称为干涉。干涉可以是建设性的或破坏性的：

1. Constructive Interference: the displacements of the two waves occur in the same direction, so the sum is an even larger wave (see Fig. 6.2, left).
2. 建设性干涉：两个波的位移发生在相同的方向上，因此和是一个更大的波（见图6.2，左）。
3. Destructive Interference: the displacements of the two waves occur in the opposite direction, so the sum is smaller than either component (see Fig. 6.2, right).
4. 破坏性干涉：两个波的位移发生在相反的方向上，因此和小于任一分量（见图6.2，右）。

Interference can be totally constructive, totally destructive, or some combination of both. The amount depends on the relative phases of the component waves - that is, the relative location of each wave in its oscillatory cycle. 干涉可以是完全建设性的，完全破坏性的，或两者的某种组合。程度取决于分量波的相对相位 - 即每个波在其振荡周期中的相对位置。

Figure 6.2: Totally constructive and totally destructive interference of two waves. 图6.2：两个波的完全建设性和完全破坏性干涉。

When visible light constructively interferes, the result is an increase in the intensity of the light. When it destructively interferes, the intensity decreases. The first clear demonstration that this actually occurs was carried out in 1801 by Thomas Young. After passing a collimated light beam through two narrow slits, Young observed the wavelike interference of the beam, which formed a pattern of bright and dark spots. 当可见光发生建设性干涉时，结果是光的强度增加。当它发生破坏性干涉时，强度减小。这种现象实际发生的第一个清晰演示是由托马斯·杨在1801年进行的。在将准直光束通过两个窄缝后，杨观察到了光束的波状干涉，形成了明暗斑点的图案。

Young's double slit experiment, illustrated in Fig. 6.3, permits what remains of an incoming wave (from the left) to travel to a distant screen (on the right) along two different paths $l_{1}$ and $l_{2}$. The light waves from the two slits interfere, resulting in an interference pattern of bright (constructive) and dim (destructive) patches as viewed on the screen. 杨氏双缝实验，如图6.3所示，允许入射波（从左侧）的剩余部分沿着两条不同的路径 $l_{1}$ 和 $l_{2}$ 传播到远处的屏幕（在右侧）。来自两个缝的光波相互干涉，在屏幕上形成明亮（建设性）和暗淡（破坏性）斑块的干涉图案。

Using the geometry of Fig. 6.3 and Fig. 6.4, we can quantitatively predict where bright and dark patches will appear on the screen. The two slits are separated by a distance $d$, and located a distance $D$ from the screen. We shine monochromatic (single-wavelength) light of wavelength $\lambda$ from the left onto the double slit, which allows two light waves to propagate from the two slits. Straight ahead, they will always be in phase because they travel the same distance to the screen. But when the two waves propagate at an angle $\theta$, they cover different distances to reach a specific point on the screen ${ }^{1}$. 使用图6.3和图6.4的几何关系，我们可以定量预测明亮和黑暗的斑点将在屏幕上出现的位置。两个缝隙之间相隔距离$d$，并位于距屏幕的距离$D$处。我们从左侧向双缝照射单色（单一波长）光，波长为$\lambda$，这使得两束光波从两个缝隙传播出去。在正前方，它们总是处于相同的相位，因为它们到达屏幕的距离相同。但当两个波以角度$\theta$传播时，它们到达屏幕上特定点的距离不同${ }^{1}$。

Figure 6.3: Interference of light emanating from two small slits. Note that the horizontal distance $D$ has been shortened and distorted in this drawing. 图6.3：从两个小缝隙发出的光的干涉。注意，在这个图示中，水平距离$D$已被缩短和扭曲。

From Fig. 6.4, it is clear that the difference in path length $\Delta l$ between the two slits is 从图6.4可以清楚地看出，两个缝隙之间的路径长度差$\Delta l$为

$$\Delta l=d \sin \theta$$

It is this $\Delta l$ that determines the location of intensity maxima and minima in the interference pattern on the screen. For an intensity minimum to occur, we must have destructive interference between the two waves. This happens when the length difference between the waves' paths is a half-integer multiple of the wavelength of the light: 正是这个$\Delta l$决定了屏幕上干涉图案中强度极大值和极小值的位置。要出现强度极小值，我们必须在两个波之间有破坏性干涉。这发生在波的路径之间的长度差是光的波长的半整数倍时：

$$\Delta l=\left(m+\frac{1}{2}\right) \lambda, \quad m=0, \pm 1, \pm 2 \ldots$$

When $\Delta l$ takes on these values, the relative phase between the two waves will be $180^{\circ}$ (consult Fig. 6.2 to convince yourself). For an intensity maximum, the relative phase must be $0^{\circ}$, and this occurs when $\Delta l$ is an integer multiple of the wavelength: 当$\Delta l$取这些值时，两个波之间的相对相位将是$180^{\circ}$（参考图6.2以确信这一点）。对于强度极大值，相对相位必须是$0^{\circ}$，这发生在$\Delta l$是波长的整数倍时：

$$\Delta l=m \lambda, \quad m=0, \pm 1, \pm 2 \ldots$$

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Figure 6.4: Geometry of the double slit, assuming that the emitted rays are effectively parallel. 图6.4：双缝的几何结构，假设发射的光线基本上是平行的。

Therefore, there exists a set of angles $\theta_{m}$ where the intensity maxima will occur, satisfying 因此，存在一组角度$\theta_{m}$，在这些角度处会出现强度极大值，满足

$$d \sin \theta_{m}=m \lambda$$

For small $\theta_{m}$ (in radians), we can use the approximation $\sin \theta \approx \tan \theta$. If $x$ is the distance on the screen from the central maximum, then the position of the $m^{\text {th }}$ maximum is given by 对于小的$\theta_{m}$（以弧度计），我们可以使用近似$\sin \theta \approx \tan \theta$。如果$x$是从中心极大值到屏幕上的距离，那么第$m^{\text {th }}$个极大值的位置由下式给出

$$\sin \theta_{m} \approx \tan \theta_{m} \approx \frac{x_{m}}{D}$$

Combining all of our results, we find that the position of the $m^{\text {th }}$ maximum is a linear function of $m$ : 综合我们所有的结果，我们发现第$m^{\text {th }}$个极大值的位置是$m$的线性函数：

$$\begin{equation*} x_{m}=\left(\frac{\lambda D}{d}\right) m \tag{6.2} \end{equation*}$$

2.3 The Interference Pattern 2.3 干涉图案

The cartoon in Fig. 6.3 suggests that the intensity pattern produced by the double slit should look roughly like a sine wave, with distance between the peaks depending on the slit separation $d$ and the screen distance $D$. While this is true, it is not the entire story. 图6.3中的示意图表明，双缝产生的强度图案应该大致看起来像一个正弦波，峰值之间的距离取决于缝隙间距$d$和屏幕距离$D$。虽然这是正确的，但这并不是全部内容。

In fact, the true intensity pattern should appear as depicted in Fig. 6.5. The double slit diffraction pattern (solid line) is periodic, but it also contains some larger features. This results from the fact that a combination of interference effects are present in the double slit experiment: first, there is interference of the two slits, as we have already discussed, and, secondly, there is a diffraction effect that occurs from each individual slit. The larger pattern in the figure is the intensity profile you would observe if you just passed the light through one slit. The intensity of the single slit pattern is given by 事实上，真正的强度图案应该如图6.5所示。双缝衍射图案（实线）是周期性的，但它还包含一些更大的特征。这是由于在双缝实验中存在干涉效应的组合：首先，正如我们已经讨论过的，有两个缝隙的干涉，其次，每个单独的缝隙都会产生衍射效应。图中较大的图案是如果你只让光通过一个缝隙时会观察到的强度分布。单缝图案的强度由下式给出

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

Figure 6.5: Intensity pattern of the double slit: a double slit pattern modulated by a single slit envelope. 图6.5：双缝的强度图案：由单缝包络调制的双缝图案。

where $a$ is the width of the slit. From this result, you can show that the single slit minima occur when 其中$a$是缝隙的宽度。从这个结果，你可以证明单缝极小值出现在

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

Using the small angle approximation we applied earlier, we can express this result in terms of the positions $x_{n}$ of the single slit minima on the viewing screen: 使用我们之前应用的小角度近似，我们可以用观察屏幕上单缝极小值的位置$x_{n}$来表达这个结果：

$$\begin{equation*} x_{n}=\left(\frac{\lambda D}{a}\right) n \tag{6.3} \end{equation*}$$

3 Experiment 3 实验

To conduct the experiment, you will use the components shown in Fig. 6.6. The primary pieces of equipment that you will need to use are: 要进行实验，您将使用图6.6中显示的组件。您需要使用的主要设备****部件是：

Light Sensor with Rotary Motion Sensor (RMS): Light sensor is a light-sensitive device that measures the total power of light incident on it, and RMS measures and reports the current transverse position. It is mounted on a linear translator, along which it can move on the direction transverse to the optical bench. 带有旋转运动传感器（RMS）的光传感器：光传感器是一种对光敏感的设备，用于测量入射在其上的光的总功率，而RMS测量并报告当前的横向位置。它安装在线性平移台上，可以沿着与光学平台横向的方向移动。

Double/Single Slit Disk: Holds various kinds of slits that will be used in this experiment. Slits can be switched by rotating the disk. 双/单缝盘：容纳将在本实验中使用的各种类型的缝隙。通过旋转圆盘可以切换缝隙。

Polarizer (with RMS): Polarizes unpolarized light or act as analyzer. The angle it rotates through can be measured and reported with a RMS. 偏振器（带RMS）：将非偏振光偏振或作为分析器。它旋转的角度可以通过RMS测量和报告。

Figure 6.6: Equipment components used in the experiment. 图6.6：实验中使用的设备组件。

During the first part of the experiment, you will use the polarizer with RMS to test Malus' Law, with the laser being the light source. In the second part of the experiment, you will remove the polarizer from the bench, install the slits and observe double and single-slit diffraction of laser light. The laser is extremely useful because it emits intense light of a single wavelength, and because this light is coherent (all light waves are in phase). 在实验的第一部分，您将使用带有RMS的偏振器来测试马吕斯定律，以激光器作为光源。在实验的第二部分，您将从平台上移除偏振器，安装缝隙并观察激光光的双缝和单缝衍射。激光器非常有用，因为它发射单一波长的强烈光，并且这种光是相干的（所有光波都处于相同的相位）。

It is very important that lasers, polarizers and disks should stand upright at all times. Laying them on their sides exposes them to damage from dust and to getting scratched/damaged by another object placed on top of it. 非常重要的是，激光器、偏振器和圆盘应始终保持直立。将它们放在侧面会使它们暴露在灰尘造成的损坏中，并可能被放在其上的其他物体刮伤/损坏。

3.1 Safety Note 安全注意事项

[]: �{ }^{1}$我们假设两条**光线**是平行的$\left(\theta_{1} \approx \theta_{2}\right)$，这是一个很好的**近似**，因为到**屏幕**的**距离**$D$与两个**缝隙**之间的**距离**$d$相比实际上是无限大的。

Although the laser is of relatively low intensity, it can be dangerous in certain circumstances unless used carefully. In particular, do not use the laser in such a way that it can shine into any person's eye. (The warning label on the laser states "Do NOT stare into beam!"). When you are not using the laser, remember to turn it off. 尽管激光器的强度相对较低，但在某些情况下如果不小心使用可能会很危险。特别是，不要以可能照射到任何人眼睛的方式使用激光器。（激光器上的警告标签写着"不要盯着光束看！"）。当您不使用激光器时，记得将其关闭。

4 Procedure 4 程序

4.1 Polarization 4.1 偏振

• Make sure only the dim incandescent ceiling lights in the room are on.
• 确保房间内只有昏暗的白炽顶灯开着。
• Place the Polarization Analyzer with the Rotary Motion Sensor (RMS) between the laser and the Light Sensor.
• 将带有旋转运动传感器（RMS）的偏振分析器放置在激光器和光传感器之间。
• Rotate the aperture disk in front of the Light Sensor so that the open aperture is in front of the light sensor. Make sure that the light from the laser is aligned properly with the aperture. Push the $0-10,000$ sensitivity button on the side of the Light Sensor.
• 旋转光传感器前面的光圈盘，使开放的光圈位于光传感器前面。确保来自激光器的光与光圈正确对齐。按下光传感器侧面的$0-10,000$灵敏度按钮。
• Push all components on the Optics Track as close together as possible.
• 将光学轨道上的所有组件尽可能地推近。
• Open "Polarization" in DataStudio. Make sure the Rotary Motion Sensor will measure the Angular Position and use Large Groove in the setup.
• 在DataStudio中打开"偏振"。确保旋转运动传感器将测量角位置并在设置中使用大槽。
• Click Start and slowly rotate the polarizer through 360 degrees (one revolution), then click Stop. Try to move slowly and steadily through the turning points.
• 点击开始并慢慢旋转偏振器360度（一圈），然后点击停止。尝试在转折点处缓慢稳定地移动。
• Using the Smart Tool record the coordinates of at least 20 data points from the Intensity vs. Angular position graph. The associated measurement uncertainties are $\pm 1$ on $0-10,000$ sensitivity, $\pm 0.01$ on $0-100$ sensitivity for the relative intensity and $\pm 0.1^{\circ}$ for the angle.
• 使用智能工具记录强度与角位置图中至少20个数据点的坐标。相关的测量不确定性在$0-10,000$灵敏度下为$\pm 1$，在$0-100$灵敏度下相对强度为$\pm 0.01$，角度为$\pm 0.1^{\circ}$。
• Make sure to record the maximum intensity, $I_{0}$, and the respective angular position, $\theta_{0}$. Maximum intensity occurs when the polarizer is aligned with the direction of the polarized laser light, so you will need to subtract $\theta_{0}$ from your measured angles to make sure you are recording the angle difference between the polarization directions of the laser and the polarizer.
• 确保记录最大强度$I_{0}$和相应的角位置$\theta_{0}$。当偏振器与偏振激光光的方向对齐时，会出现最大强度，因此您需要从测量的角度中减去$\theta_{0}$，以确保您记录的是激光器和偏振器的偏振方向之间的角度差。

4.2 Double Slit Diffraction Pattern 4.2 双缝衍射图案

• Replace the USB connector of the RMS on the Polarizer with that of the RMS on the Light Sensor.
• 将偏振器上的RMS的USB连接器替换为光传感器上的RMS的连接器。
• Set the Double Slit disk to the ( $a=0.04 \mathrm{~mm}, d=0.25 \mathrm{~mm}$ ) position. Here $a$ refers to the slits' width and $d$ to the separation between the two slits.
• 将双缝盘设置到（$a=0.04 \mathrm{~mm}, d=0.25 \mathrm{~mm}$）位置。这里$a$指的是缝隙宽度，$d$指的是两个缝隙之间的间隔。
• Place the laser at the far end of the Optics Track. Record the wavelength of the laser as printed on its back. Mount the High Precision Double Slit Disk to the optics bench next to the laser, with the printed side toward the laser.
• 将激光器放在光学轨道的远端。记录印在激光器背面的波长。将高精度双缝盘安装在激光器旁边的光学平台上，印刷的一侧朝向激光器。
• Measure and record the distance $D$ from the position of the double slit to the tip of the Light Sensor. You can use either the scale on the track or use a meter stick.
• 测量并记录从双缝的位置到光传感器的尖端的距离$D$。您可以使用轨道上的刻度或使用米尺。
• Move the Light Sensor slightly left or right along the linear translator until you can see the beam somewhere on the white screen. Use the adjustment screws on the back of the laser to adjust the position of the laser beam from left-to-right and up-and-down to make the pattern on the white screen as bright as possible and aligned with the height of the slits on the Light Sensor.
• 沿着线性平移器稍微向左或向右移动光传感器，直到您可以在白色屏幕上的某处看到光束。使用激光器背面的调节螺丝从左到右和上下调整激光束的位置，使白色屏幕上的图案尽可能明亮，并与光传感器上的缝隙的高度对齐。
• Use slit #3 on the Light Sensor and push the 0-100 sensitivity button.
• 使用光传感器上的缝隙#3并按下0-100灵敏度按钮。
• Move the Light Sensor to the left end of the linear translator arm as viewed from the laser.
• 从激光器看，将光传感器移动到线性平移臂的左端。
• Open "Interference" in DataStudio. Click Start.
• 在DataStudio中打开"干涉"。点击开始。
• Slowly move the Light Sensor across the translator arm and scan the full pattern. Hold the rear of the RMS down against the linear translator bracket so it does not wobble up and down as it moves. Try to move the sensor as smoothly as possible.
• 慢慢地将光传感器横跨平移臂移动并扫描完整的图案。将RMS的后部按在线性平移支架上，使其在移动时不会上下摇晃。尝试尽可能平稳地移动传感器。
• Click Stop when you have finished scanning the light pattern and reached the right end of the linear translator.
• 当您完成光图案的扫描并到达线性平移器的右端时，点击停止。
• Use the Smart Tool to record the positions of the maxima on the central interference pattern.
• 使用智能工具记录中央干涉图案上的极大值的位置。

4.3 Single Slit Envelope 4.3 单缝包络

• Please keep the data from the double-slit part. You can choose to hide the previous data in DataStudio so that the new data look clear.
• 请保留双缝部分的数据。您可以选择在DataStudio中隐藏先前的数据，使新的数据看起来清晰。
• Move the Light Sensor to the center of the linear translator arm so that you can see the interference pattern on the white screen.
• 将光传感器移动到线性平移臂的中心，以便您可以在白色屏幕上看到干涉图案。
• Swap out the Double Slit Disk for the Single Slit Disk. Set the disk to the ( $a=0.04 m m$ ) position, and check that the optics are aligned so that you can see a clear single slit pattern on the white screen. See Fig. 6.7.
• 将双缝盘换成单缝盘。将圆盘设置到（$a=0.04 m m$）位置，并检查光学元件是否对齐，以便您可以在白色屏幕上看到清晰的单缝图案。参见图6.7。
• To make better measurements, you will probably want to use a different slit on the light sensor. Try slit #5 for example. Make sure to take note of any changes made.
• 为了进行更好的测量，您可能想要在光传感器上使用不同的缝隙。例如尝试缝隙#5。确保记录所做的任何更改。
• Move the Light Sensor to the left end of the linear translator arm as viewed from the laser.
• 从激光器看，将光传感器移动到线性平移臂的左端。
• Click start in DataStudio. You can now record the positions of the minima of the single slit pattern at the same distance $D$. Remember to click stop.
• 在DataStudio中点击开始。您现在可以记录相同距离$D$处的单缝图案的极小值的位置。记得点击停止。
• Take a screenshot of the overlapping curves.
• 拍摄重叠曲线的截图。

Figure 6.7: Single slit pattern on the white screen. 图6.7：白色屏幕上的单缝图案。

5 Analysis 5 分析

Polarization 偏振

To test Malus' Law, plot $I / I_{0}$ against $\cos ^{2} \theta$, where $\theta$ is the angle difference between the polarizer's angular position and $\theta_{0}$. Does this look like a straight line, as you would expect from Malus' Law? Perform a regression analysis and record the slope and intercept of the line, with their uncertainties. You can also choose to use some non-linear fitting tools to fit your original data with the $\cos ^{2}$ function, and check the quality of the fit. In addition, consider the following questions: 为了测试马吕斯定律，绘制$I / I_{0}$对$\cos ^{2} \theta$的图，其中$\theta$是偏振器角位置与$\theta_{0}$之间的角度差。这看起来像一条直线吗，正如您从马吕斯定律中所期望的那样？执行回归分析并记录线的斜率和截距，以及它们的不确定性。您也可以选择使用一些非线性拟合工具用$\cos ^{2}$函数拟合您的原始数据，并检查拟合的质量。此外，考虑以下问题：

• What reading do you obtain when the polarizer is at an angle of $90^{\circ}$ relative to where maximum intensity is recorded? This is called the noise of your measuring device. What reading would you expect if there was no noise?
• 当偏振器处于相对于记录最大强度的位置$90^{\circ}$的角度时，您获得什么读数？这被称为测量设备的噪声。如果没有噪声，您期望什么读数？
• What is the signal to noise ratio? The signal is the reading at $\theta_{0}$. What does this tell you?
• 什么是信号与噪声比？信号是在$\theta_{0}$处的读数。这告诉您什么？
• If you had a lot of background light, how could you reduce the influence it would have on your results: by changing the setting, or by changing your data analysis?
• 如果您有很多背景光，您如何减少它对您的结果的影响：通过改变设置，还是通过改变您的数据分析？
• How does the noise arise? Is it possible to eliminate it completely?
• 噪声是如何产生的？是否可能完全消除它？
• What is the physical meaning of the $y$-intercept of your plot? If it is different from zero, is the difference statistically significant?
• 您的图的$y$-截距的物理意义是什么？如果它与零不同，这个差异在统计上是否显著？
• What results would you get if you performed the same experiment using the incandescent bulb as a light source, instead of the laser?
• 如果您使用白炽灯泡作为光源而不是激光器执行相同的实验，您会得到什么结果？

Summary of data: 数据摘要：

• Polarization section:
• 偏振部分：
• Maximum intensity $I_{0}$ and the corresponding position $\theta_{0}$
• 最大强度$I_{0}$和相应的位置$\theta_{0}$
• Variation in the intensity with polarizer angle, 20 pairs
• 偏振器角度的强度变化，20个数据对
• Double slit diffraction section:
• 双缝衍射部分：
• Wavelength of the laser
• 激光器的波长
• Distance $D$ from slits to linear translator
• 从缝隙到线性平移器的距离$D$
• Slit separation $d$
• 缝隙间距$d$
• Positions of maxima, $x_{m}$
• 极大值的位置，$x_{m}$
• Single slit envelope
• 单缝包络
• Distance $D$ from slit to linear translator
• 从缝隙到线性平移器的距离$D$
• Slit width, $a$
• 缝隙宽度，$a$
• Positions of minima
• 极小值的位置

Double Slit Diffraction Pattern 双缝衍射图案

• Begin analyzing your double slit data by plotting the positions of the double slit diffraction maxima $x_{m}$ against the order number $m$.
• 通过绘制双缝衍射极大值的位置$x_{m}$对级数$m$的图开始分析您的双缝数据。
• Perform a least squares analysis to find the slope of the data (with uncertainties), and use the slope to estimate the wavelength $\lambda$ of the laser light.
• 执行最小二乘分析以找到数据的斜率（带有不确定性），并使用斜率估计激光光的波长$\lambda$。
• Compare your result to the wavelength of the laser recorded before.
• 将您的结果与之前记录的激光器的波长进行比较。
• How many fringes can you see? Why can't you see more? How could you improve the experiment to see more fringes?
• 您能看到多少条纹？为什么您看不到更多？您如何改进实验以看到更多条纹？
• What limits the precision of this measurement?
• 什么限制了这个测量的精度？
• Does it matter how close the slits are to the laser? What effect does changing the slit-to-light sensor spacing have?
• 缝隙与激光器的距离近远是否重要？改变缝隙到光传感器的间距有什么影响？

Single Slit Envelope 单缝包络

• Create a plot of $x_{n}$ versus the order number $n$ (which counts up in either direction from the central maximum).
• 创建$x_{n}$对级数$n$的图（从中心极大值向任一方向计数）。
• Determine the slope of this plot, and use it and the value of $\lambda$ you found above to estimate the slit widths $a$. Do the widths agree with the nominal width written on the component?
• 确定这个图的斜率，并使用它和您上面找到的$\lambda$的值来估计缝隙宽度$a$。这些宽度是否与组件上标注的标称宽度一致？
• How big is the double slit envelope compared to the single slit envelope? What does this mean for the relative sizes of the slit width and slit separation? What happens if we change $a$ of the double-slit?
• 双缝包络与单缝包络相比有多大？这对缝隙宽度和缝隙间距的相对尺寸意味着什么？如果我们改变双缝的$a$会发生什么？

[^0]: ${ }^{1}$ We assume that the two rays are parallel $\left(\theta_{1} \approx \theta_{2}\right)$, a good approximation since the distance to the screen $D$ is effectively infinite compared to the distance $d$ between the two slits.