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Chapter15  第15章

Optics 光学

Optics is the branch of physics which deals with the behavior and properties of light, in, cluding its interactions with matter and the construction of instruments. In general, the term "light" in physics refers to electromagnetic radiation of any wavelength, whether visible of not. In this chapter, "light" is visible light that is visible to the human eye and electromagnetic radiation with a wavelength in a range from about to about , with a frequency range of about to .
光学是物理学的一个分支,它处理光的行为和特性,包括光与物质的相互作用和仪器的构造。一般来说,物理学中的“光”一词是指任何波长的电磁辐射,无论是否可见。在本章中,“光”是人眼可见的可见光和波长在大约 到大约 ,频率范围大约 到的 电磁辐射。
One of our principal contacts with the world around us is through light. We personally depend on light to convey visual information and have many instruments which use light to op. erate such as microscope, telescope and spectrometer. Most of what we know about the Universe comes from information that has been carried to us by light. Our knowledge on the structure of atoms comes largely from observing the radiations they emit. Therefore, light is very important to us, and nothing is visible to humans when light is totally absent.
The history of elucidating the nature of light can be traced back to the beginning of human being. In ancient China, India and Greek, "light" is one of the fundamental "subtle" elements such as Water, Wood, Metal, Fire, and Earth which compose of everything (that should not be confused with the ordinary meaning of these terms). In about , Euclid studied the properties of light. Euclid postulated that light travelled in straight lines and he described the laws of reflection and studied them mathematically. René Descartes (15961650) held that light was a mechanical property of the luminous body and assumed that light behaved like a wave.
阐明光的本质的历史可以追溯到人类的起源。在古代中国、印度和希腊语中,“光”是构成万物的基本“微妙”元素之一,如水、木、金、火和土(不应与这些术语的普通含义混淆)。在大约 ,欧几里得研究了光的性质。欧几里得假设光是直线传播的,他描述了反射定律并用数学方法研究了它们。勒内·笛卡尔(René Descartes,15961650)认为光是发光体的机械特性,并假设光的行为类似于波。
Isaac Newton stated in 1675 that light was composed of corpuscles (particles of matter) which were emitted in all directions from a source. Newton's theory could be used to predict the reflection of light, but could only explain refraction by incorrectly assuming that light accelerated upon entering a denser medium because the gravitational pull was greater. Although there were some outstanding questions, Newton's view was accepted by many scientists and popularized by his followers for over 100 years.
艾萨克·牛顿(Isaac Newton)在1675年指出,光是由微粒(物质粒子)组成的,这些微粒从源头向各个方向发射。牛顿的理论可以用来预测光的反射,但只能通过错误地假设光在进入密度较大的介质时加速来解释折射,因为引力更大。尽管存在一些悬而未决的问题,但牛顿的观点被许多科学家所接受,并被他的追随者推广了 100 多年。
Christiaan Huygens in 1678 proposed that light was emitted in all directions as a series of waves in a medium. As waves are not affected by gravity, it was assumed that they slowed down upon entering a denser medium. The wave theory predicted that light waves could interfere with each other like sound waves, and that light could be polarized, if it were a transverse wave. Early in the 19th century, Young showed that light behaved as waves by means of his famous double-slit diffraction experiment. Young's experiments supported the theory that
克里斯蒂安·惠更斯(Christiaan Huygens)在1678年提出,光在介质中以一系列波的形式向各个方向发射。由于波不受重力的影响,因此假设它们在进入密度较大的介质时会减速。波动理论预言,光波可以像声波一样相互干扰,如果光是横波,光可以偏振。早在19世纪,Young通过他著名的双缝衍射实验证明了光表现为波。Young的实验支持了以下理论:

light consists of waves. A subsequent work by Fresnel soon convinced scientists that light must be a wave, who independently worked out his own wave theory of light. In 1862, James
Clerk Maxwell came up with some brilliant equations, today known as Maxwell's equations and concluded that light was a form of electromagnetic radiation. His theory was later experimentally proven correct by Heinrich Hertz. The weakness of the wave theory was that light waves, like sound waves, would need a medium for transmission. Hence, "ether" (the medium) was invented. However, Michelson-Morley experiments demonstrated that ether does not exist.
书记员麦克斯韦提出了一些绝妙的方程,今天被称为麦克斯韦方程,并得出结论,光是电磁辐射的一种形式。他的理论后来被海因里希·赫兹(Heinrich Hertz)实验证明是正确的。波动理论的弱点是光波和声波一样,需要一种介质来传输。因此,“以太”(媒介)被发明了。然而,迈克尔逊-莫雷实验表明,以太并不存在。
Max Planck in 1900 suggested the idea that black bodies emit light (and other electromagnetic radiation) only as discrete bundles or packets of energy (quanta) and accurately predicted the results of "black-body radiation" observed. Albert Einstein in 1905 developed the particle theory of black-body radiation to explain the photoelectric effect. He suggested that the quantization used by Planck reflects a basic aspect of the reality and light exhibits as a particle. Later, the particle of light was given the name photon with an energy and a momentum P. His ideas were experimentally confirmed by American scientist Robert Millikan a year later. However, his ideas were finally accepted throughout the world only a decade later when the American physicist Compton made theoretical predictions for the scattering of photons from the electron-Compton effect.
马克斯·普朗克(Max Planck)在1900年提出了黑体仅以离散的能量束或能量包(量子)的形式发射光(和其他电磁辐射)的想法,并准确地预测了观察到的“黑体辐射”的结果。阿尔伯特·爱因斯坦(Albert Einstein)在1905年发展了黑体辐射的粒子理论来解释光电效应。他认为,普朗克使用的量子化反映了现实的一个基本方面,而光则以粒子的形式表现出来。后来,光粒子被命名为光子,具有能量 和动量P。一年后,美国科学家罗伯特·米利坎(Robert Millikan)通过实验证实了他的想法。然而,仅仅十年后,当美国物理学家康普顿对电子康普顿效应的光子散射做出理论预测时,他的想法才最终被全世界接受。
Louis de Broglie in 1924 stated that everything has both a particle nature and a wave nature as light. In a word, the modern theory explains the nature of light as wave-particle duality, that is light sometimes exhibits a particle nature and sometimes a wave nature. The quantum theory of light and electromagnetic radiation continued to evolve through the 1920s and 1930s, and culminated with the development during the 1940s of the theory of quantum electrodynamics (QED).
路易斯·德布罗意(Louis de Broglie)在1924年指出,万物都具有粒子性质和光的波动性质。总之,现代理论将光的本质解释为波粒二象性,即光有时表现出粒子性质,有时表现出波动性质。光和电磁辐射的量子理论在 1920 年代和 1930 年代继续发展,并在 1940 年代量子电动力学 (QED) 理论的发展中达到顶峰。
In this chapter, we briefly introduce the geometrical optics and its applications, and use the knowledge of wave motion in chapters 13 and 14 to treat interference, diffraction, and polarization of light.

15. 1 Laws of Reflection and Refraction
15. 1 反射和折射定律

15.1.1 The ray approximation in geometric optics
15.1.1 几何光学中的射线近似

Geometrical optics describes light propagation in terms of "rays". A beam of light from its source, on common experience, travels in a straight-line path at a finite speed in a homogeneous medium until it counters a boundary between two different materials. When a beam of light strikes a boundary, it is reflected from that boundary or passes into the material on the other side of the boundary or partially does both. These phenomena are reflection and refraction. The ray approximation is used to represent beams of light. That is a ray of light is an imaginary line drawn along the direction of travel of the light beam. The "ray" is used to approximately model how light will propagate. Light rays are defined to propagate in a rectiline-

ar path as far as they travel in a homogeneous medium. Rays bend (and may split in ) the interface between two dissimilar media. A light ray is a line or curve that is perpendicular
AR路径,只要它们在均匀介质中传播。光线弯曲(并可能分裂 )两种不同介质之间的界面。光线是一条垂直的线或曲线

to the light's wavefronts.
Light is the fastest thing observed in universe. The speed of light in a vacuum is exactly , and approximately , labeled as , a physical constant. When light enters a material, it slows down. For example, light travels about slower in water than it does in a vacuum, while in diamonds, it travels at about half the speed it does in a vacuum. The index of refraction of medium is introduced to measure the speed of light in that medium, defined as
光是宇宙中观测到的最快的东西。真空中的光速正好 是 ,并且近似 于 ,标记为 ,一个物理常数。当光线进入材料时,它会减慢速度。例如,光在水中的传播 速度比在真空中慢,而在钻石中,它的传播速度大约是真空中的一半。引入介质的折射率来测量该介质中的光速,定义为
where is the speed of light in the material. Eq. (15-1) implies that the index of vacuum is taken as unity " 1 ", and the index of refraction of air differs little from unity and in most situations it is considered to be unity without significant difference. As an example, the index of refraction of water is 1.333 , meaning that light travels 1.333 times faster in a vacuum than it does in water. Index of refraction of medium 2 relative to medium 1 is equal to the ratio of the speed of light in the first medium to the speed of light in the second medium
其中 是材料中的光速。式(15-1)意味着真空的折射率被取为单位“1”,空气的折射率与单位差别不大,在大多数情况下被认为是单位,没有显著差异。例如,水的折射率为 1.333 ,这意味着光在真空中的传播速度是水中的 1.333 倍。介质 2 相对于介质 1 的折射率等于第一介质 中的光速与第二介质 中的光速之比

15. 1.2 Laws of reflection and refraction
15. 1.2 反射和折射定律

When light goes from one transparent medium such as water or glass to another with different optical properties, usually there is a reflected bean as well as a refracted one as shown in Fig. 15-1, in which the ray 1 represents the incident light, the ray 2 reflected light and the ray 3 refracted light. The incident beam of light in the medium of refractive index encounters the interface (assuming to be a plane) between two media. Part of the light is reflected by the interface to form the ray 2 (reflected light), traveling as if the original beam had bounced from the interface. The rest of the light passes through the interface and into the medium of refractive index , forming the ray 3 (refracted light). The travel of light through an inter face between two media is called refraction. Unless an incident beam of light is perpendicular to the interface, the directions of reflected light and refracted light have some changes with respect to the direction of incident light. For this reason, the beam is said to be bent by refraction.
当光从一种透明介质(如水或玻璃)传播到另一种具有不同光学特性的介质时,通常有反射豆和折射豆,如图15-1所示,其中射线1代表入射光,射线2代表反射光,射线3代表折射光。折射率 介质中的入射光束遇到两个介质之间的界面(假设是一个平面)。部分光被界面反射形成射线 2(反射光),传播时就像原始光束从界面反射一样。其余的光穿过界面进入折射率 介质,形成射线3(折射光)。光通过两种介质之间的界面的传播称为折射。除非入射光束垂直于界面,否则反射光和折射光的方向相对于入射光的方向会有一些变化。出于这个原因,据说光束因折射而弯曲。
The dashed line is the normal, a line perpendicular to the surface at the point where the ray hits. The plane containing the incident ray and the normal is called the plane of incidence, which is in the plane of the page in Fig. 15-1. The angle between the normal and incident is called the angle of incidence, the angle between the normal and reflected ray is the angle of reflection and the angle of between the normal and refracted ray is the angle of refraction. These angles satisfy law of reflection and law of refraction as follow.
虚线是法线,是一条垂直于光线照射点表面的线。包含入射光线和法线的平面称为入射平面,位于图 15-1 页面的平面内。法线和入射 线之间的角度 称为入射角,法线和反射线之间的夹角 是反射角,法线和折射线之间的夹角 是折射角。这些角度满足反射定律和折射定律,如下所示。
Law of reflection: (1) The angle of reflection is equal to the angle of incidence. (2) The in cident ray, the reflected ray, and the normal lie in the plane of incidence.
反射定律:(1)反射角等于入射角。(2)入 射光线、反射光线和法线位于入射平面。
Law of refraction: (1) The incident ray, the refracted ray, and the normal lie in the plane of incidence. (2) The angle of refraction is related to the angle of incidence by
Law of refraction is called Snell's law. Eq. (15-3) can be rewritten as
Eq. (15-4) implies that for a given angle of incidence, the value of depends on the relative refraction index
方程(15-4)意味着对于给定的入射角,入射角 值取决于相对折射率
Fig. 15-1 Schematic diagram of reflection and refraction of light . For example, if (that is ), is less than , and this means that refraction bends the light beam away from the undeflected direction and toward the normal.
图15-1 光 的反射和折射示意图 .例如,如果 (即 ), 则小于 ,这意味着折射使光束偏离未偏转方向并朝向法线。
If is less than is greater than . In this case, refraction bends the light beam away from the undeflected direction and away from the normal. For the case, if the angle of incidence is greater than the critical angle
如果 小于 大于 。在这种情况下,折射会使光束偏离未偏转的方向并远离法线。对于这种情况,如果入射角大于临界角
for which the angle of refraction is , there is no refracted light and all the light is reflected. This effect is called total reflection. Total reflection has found many applications such as in optical fiber communications and in medical technology.
对于折射角为 ,没有折射光,所有的光都被反射。这种效应称为全反射。全反射在光纤通信和医疗技术等领域得到了许多应用。
There are many application examples of the refraction and the total reflection in everyday life. For example, one can see a fish moving in water near to lakeside and find that the apparent depth of the fish in water is less than the actual depth as shown in Fig. 15-2. Moreover, one cannot find the fish far from lakeside. The law of refraction and total reflection can explain the daily experience.
折射和全反射在日常生活中有许多应用实例。例如,可以看到一条鱼在靠近湖边的水中移动,发现鱼在水中的表观深度小于实际深度,如图 15-2 所示。此外,人们在远离湖边的地方找不到鱼。折射定律和全反射定律可以解释日常经验。
Fig. 15-2 Example of total reflection
图15-2 全反射示例
The laws of reflection and refraction can be used to predict the deflection of light rays and the formations of images for mirrors and lenses.
Example 15-1 Find the critical angle of total reflection for a water-air boundary if the index of reflection of water is 1.33 .
例 15-1 求水-空气边界的全反射临界角,如果水的反射率为 1.33.
Solution From Eq. (15-5), we find the critical angle to be
解 从方程(15-5)中,我们发现临界角为

15. 1.3 Chromatic dispersion
15. 1.3 色散

The careful measurements have shown that the index of refraction in anything but vacuum

nds on the wavelength of light. The dependence of the index of refraction on wavelughe depends on the wavelength of light. The dependence of the index of refraction on wavelengum

indicates that when light is refracted by a surface of material, the angle of refraction depends

on the wavelength of light. This implies that a very narrow beam of white light will be on the wavelength of light. This implies that a very narrow beam of white light will be spreads out by the refraction such as passing through a prism as shown in Fig. 15-3. This separation of white light into its component colors is called chromatic dispersion. The index of refraction for a material usually decreases with increasing wavelength so that violet light refracts more for
在光的波长上。这意味着一束非常窄的白光将出现在光的波长上。这意味着一束非常窄的白光将通过折射(例如通过棱镜)散开,如图 15-3 所示。这种将白光分离成其成分颜色的现象称为色散。材料的折射率通常随着波长的增加而降低,因此紫光的折射率更高

red light when passing from air into a material.
Fig. 15-3 Schematic diagram of chromatic dispersion
图15-3 色散示意图
The dispersion of light is demonstrated most vividly in nature through the formation of a rainbow, often seen by an observer positioned between the Sun and a rain shower.

15. 2 The Plane Mirror and Spherical Mirrors
15. 2 平面镜和球面镜

A mirror is a surface that can reflect a beam of light in one direction instead of either scattering it widely in many directions or absorbing it. When light falls on a rough, opaque surface, the incident light is scattered in all directions. If the surface is so smooth that there is little random scattering, the surface is said to be polished and can be used as a mirror. In the following, we use the law of reflection to study the formation of images of an object reflected by the plane mirror, the concave spherical mirror and the convex spherical mirror.
A point on the surface of an object can be considered as a light source. Generally a source of light can be considered a point source if the resolution of the imaging instrument is too low to resolve its size, or if the object is at a very great distance.

15.2.1 Plane mirror 15.2.1 平面镜

A plane mirror is a mirror with a planar reflective surface. For light rays striking a plane mirror, the angle of reflection equals the angle of incidence. This implies that images of object are upright and the same distance behind the mirror as in front of the mirror. The image size is the same as the object size. That is the magnification of a flat mirror is equal to one. Images are classified as real or virtual. A real image is one in which light actually passes through the

image point and a virtual image is one in which the light does not pass through the image point but appears to come (diverge) from that point.
As shown in Fig. 15-4 (a), when a point object is placed in frond of a plane mirror and sends light in all directions, some of rays are reflected. To an observer in front of the mirror all the reflected rays appear to come from the point I behind the mirror. The observer therefore sees a bright spot which appears to be behind the mirror and which we call the image of the point . The image behind the mirror is a virtual image because the light rays do not actually come from . The distance is the object distance and the distance is the image distance.
如图15-4(a)所示,当将点物体 放置在平面镜的叶子中 并向各个方向发送光时,部分光线被反射。对于镜子前的观察者来说,所有反射的光线似乎都来自镜子后面的点 I。因此,观察者会看到一个亮点,它似乎在镜子后面,我们称之为点 的图像。镜子后面的图像是虚拟图像,因为光线实际上不是来自 。距离 是物体距离,距离 是图像距离。
The image of the object with height of can be found by graphical ray tracing as shown in Fig. 15-4 (b). Two rays start from point and then reflect from mirror. One follows a horizontal path to the mirror, and reflects back on itself. The second ray follows the oblique path and reflects. An observer to the left of the mirror would trace the two reflected rays back to the point from which they appear to have originated, that is the image of point . Because triangles and are identical, the image is far behind the mirror as the object in front, and the object height equals the image height . Lateral magnification is defined as follows
高度为 的 物体 的图像可以通过图形光线追踪找到,如图 15-4 (b) 所示。两条光线从点 开始,然后从镜子反射。一个人沿着一条水平路径 来到镜子前,并反射回自己。第二条光线沿着倾斜路径 反射。镜子左侧的观察者将追踪两条反射光线,追溯到它们似乎起源的点 ,即点 的图像。因为三角形 是相同的,所以图像远远落后于镜子,就像前面的物体一样,物体高度 等于图像高度 。横向放大倍率 定义如下
For a plane mirror, because .
对于平面镜, 因为 .
Fig. 15-4 Formation of image by reflection from a plane mirror
图15-4 平面镜反射图像的形成
However, the law of reflection also implies that the image is a laterally-inverted "mirror image" of the object. If a person is reflected in a plane mirror, the image of his right hand appears to be the left hand of the image.

15.2.2 Spherical mirrors
15.2.2 球面镜

A curved mirror is a mirror with a curved reflective surface, which may be either convex or concave. Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are sometimes used in optical devices.
A concave spherical mirror is part of a spherical shell with its inner surface polished as shown in Fig. 15-5 (a). The mirror has radius of curvature , and its center of curvature is at
凹面球面镜是球面壳的一部分,其内表面经过抛光处理,如图15-5(a)所示。镜子有曲率半径 ,它的曲率中心在

point . Point (vertex) is the center of the spherical segment, and a line drawn from is called the principal axis of the mirror. When the rays parallel to the principal axis reach, concave mirror, those near the central axis are reflected through a common point which is called the focal point (or focus) of the mirror (a real focal point) and its distance from the cen. ter of the mirror is the focal length of the mirror. That is
。点 (顶点)是球面线段的中心,从 中绘制的线称为镜子的主轴。当平行于主轴的光线到达凹面镜时,靠近中心轴的光线通过一个公共点反射,该公共点 称为镜子的焦点(或焦点)(真正的焦点),它与镜子的厘米的距离是镜子的焦距。那是
If an incident ray of light passes through the focus , it is reflected back in the direction of principal axis.
如果入射光线穿过焦点 ,它会在主轴方向上反射回来。
Fig. 15-5 (a) The concave mirror and (b) formation of image
图15-5 (a)凹面镜和(b)图像的形成
As shown in Fig. 15-5(b), an object is placed before a concave mirror, we can locate an image by graphical ray tracing. The ray parallel to the principal axis is reflected back through , and the ray strikes the mirror at point and is reflected back parallel to the principal axis. The ray comes to the spherical surface along the radius and is reflected directly back on itself. The is the image of object and it is a real image since the rays of light come from the . In this case, the image is inverted. Let and denote the distances of object and image respectively. They are related by a simple formula which can be derived easily
如图15-5(b)所示,将物体 放置在凹面镜之前,我们可以通过图形光线追踪来定位图像。平行于主轴的光线 被反射 回来,光线在点 处照射到镜子上,并平行于主轴反射回来。光线 沿半径到达球面,并直接反射回自身。这是 物体 的图像,它是真实的图像,因为光线来自 .在这种情况下,图像是反转的。让 分别表示物体和图像的距离。它们通过一个简单的公式相关联,可以很容易地推导出来
This equation applies to any concave, convex, or plane. For a convex or plane mirror, only a virtual image can be formed, regardless of the object's location on the central axis. For a concave mirror, different image types can be formed and depend on the distance between the object and the mirror. Note, according to Eq. (15-8), that an object and it's image are conjugate.
For a convex mirror, we find that the parallel rays are no longer reflected through a common point. Instead, they diverge. As shown in Fig. 15-6, the convex mirror is a diverging mir ror because rays initially parallel diverge after reflection. A ray of light parallel to the principal axis is reflected away from the principal axis, and the reflected ray is extended backward to
对于凸面镜,我们发现平行光线不再通过公共点反射。相反,他们分道扬镳。如图15-6所示,凸面镜是一个发散的mir ror,因为最初平行的光线在反射后会发散。平行于主轴的光线被反射到远离主轴的地方,反射的光线向后延伸至

cross the principal axis as point , the principal focus for this convex mirror. The principal focus is virtual. If we place an object , an erect virtual image is formed and reduce in size. The image on a convex mirror is always virtual, diminished, and upright. Eq. (15-8) can be used for both concave and convex mirrors if we are careful to use the proper signs for various quantities shown in Table 15-1.
以主轴为点 ,是这个凸面镜的主要焦点。主要关注点是虚拟的。如果我们放置一个物体 ,就会形成一个直立的虚拟图像 并缩小尺寸。凸面镜上的图像始终是虚拟的、缩小的和直立的。方程(15-8)既可以用于凹面镜,也可以用于凸面镜,如果我们小心地对表15-1中所示的各种量使用适当的符号。
Fig. 15-6 The convex mirror and formation of image
图15-6 凸面镜与图像的形成
Table 15-1 Sign conventions for mirrors
表 15-1 镜像签名约定
Quantity 数量 Positive when  Negative when 阴性时
Object location
Object is in front of Mirror
Object is behind mirror
Image location  图像位置 Image is behind mirror
Image is in front of Mirror
Image height  图像高度 Image is upright 图像是直立的 Image is inverted 图像反转
Radius (or focal length )
半径 (或焦距
Mirror is concave 镜子是凹面的 Mirror is convex 镜子是凸的
The lateral magnification can be proved to be
By convention, the lateral magnification includes a plus sign when the image orientation is that of the object and a minus sign when the orientation is opposite that of the object. The imageforming characteristics of curved mirrors obviously determine their uses.
Example 15-2 An object of height is placed from a convex mirror with a focal length of . Find
例 15-2 从焦距为 的 凸面镜放置 高度 物体。找到
(1) the position of the formed image;
(2) the magnification of the mirror.
Solution (1) Because the mirror is convex, its focal length is negative. According to Eq. (15-8), we have
The negative value of indicates that the image is virtual and behind the mirror.
负值表示 图像是虚拟的,并且在镜子后面。
(2) The magnification of mirror is
Hence, the image is upright.

15.3 Images Formed by Refraction and Thin Lenses
15.3 折射和薄透镜形成的图像

15.3.1 Images formed by refraction
15.3.1 折射形成的图像

We now turn from images formed by reflections through spherical surfaces of transparent materials, such as glass. Consider two transparent media with indices of refraction and , where the
现在,我们从反射形成的图像转向透明材料(如玻璃)的球面。考虑两个具有折射率 的透明介质,其中

boundary between the two media is a spherical surface of radius . At the point of refraction of each ray, the normal to the refracting surface is a radial line through the center of curvature. Because of refraction, the ray bends toward the normal if it is entering a medium of greater index of refraction, and away from the normal if it is entering a medium of lesser index of refraction.
两种介质之间的边界是半径为球面 的球面。在每条光线的折射点,折射面的法线是一条穿过曲率中心的径向线。由于折射,如果光线进入折射率较大的介质,则光线会向法线弯曲,如果光线进入折射率较低的介质,则光线会偏离法线。
Fig. 15-7 An Image formed by refraction through a spherical surface
图15-7 通过球面折射形成的图像
Consider a ray from the point source in medium 1 which is incident upon a spherical surface (convex) of medium 2 with center of curvature at as shown in Fig. 15-7. A line drawn from to the center of the segment of spherical surface is called the principal axis of the system. Ray is refracted at surface and an image is formed. According to law of refraction, we have
考虑来自介质 1 中点源 的光线,该光线 入射到介质 2 的球面(凸面),曲率中心如 图 15-7 所示。从 球面段中心 绘制的线称为系统的主轴。光线 在表面折射并形成图像。根据折射定律,我们有
It is easy to prove that
For the paraxial rays, i. e., rays which make small angles with the axis, their sine, their tangent and themselves are essentially equal. Hence we have
Then, by Eq. (15-10), we have
Note that real images are formed on the side of surface opposite the side from which the light comes. This is in contrast with mirrors. Eq. (15-12) applies to a variety of circumstances: concave convex and plane surfaces ( approaches infinity). The sign convention for spherical refracting surface is summarized in Table 15-2.
请注意,真实图像是在与光线来自的一侧相对的表面一侧形成的。这与镜子形成鲜明对比。方程(15-12)适用于各种情况:凹面凸面和平面( 接近无穷大)。球面折射面的符号约定总结于表15-2中。
Table 15-2 Sign conventions for refracting surface
表15-2 折射面的符号约定
Quantity 数量 Positive when  Negative when 阴性时
Object location
Object is in front of surface
Object is in back of surface
Image location  图像位置 Image is in back of surface
Image is in front of surface
Image height  图像高度 Image is upright 图像是直立的 Image is inverted 图像反转
Radius  半径 Center of curvature is in back of surface
Center of curvature is in front of surface
Furthermore, the lateral magnification of a refracting surface is

15.3.2 Thin lenses 15.3.2 薄镜头

Our knowledge of how light rays are bent when they pass from one material to another can now be applied to lenses, which are the basic elements of most optical instruments. A lens is a transparent object with two refracting surfaces whose central axes coincide. The common central axis is the principal axis of the lens. Simple lenses are bounded by faces which are small sections of sphere. When a lens is surrounded by air, light refracts from the air into the lens, crosses through the lens, and then refracts back into the air. Each refraction can change the direction of travel of the light. A lens that causes light rays initially parallel to the central axis to converge is called a converging lens. If, instead, it causes such rays to diverge, the lens is a diverging lens. Fig. 15-8 shows six possible types of spherical lenses with their name: double-convex, plano-convex, plano-concave, concavo-convex, convexo-concave, and doubleconcave.
Fig. 15-8 Six possible types of spherical lenses
图15-8 球面透镜的六种可能类型
A thin lens is a lens in which the thickest part is thin compared to the object distance , the image distance and the radii of curvature and of the two surfaces of lens. The thin lens approximation ignores optical effects due to the thickness of lenses and simplifies ray tracing calculations. Rays which approach a converging lens parallel to the principal axis are deviated so that they pass through a common point, the principal focus, on the principal axis. A lens has two principal foci, one for light incident from the left, which we denote by , and the second for light incident from the right. For thin lenses the principal foci are equidistant from the optical center of the lens as shown in Fig. 15-9. The focal length of a thin lens with index of refraction surrounded by air is given by the Lensmaker's equation
薄透镜是指与物距 、像距 、曲率 半径和 透镜的两个表面相比,最厚部分较薄的透镜。薄透镜近似忽略了由于透镜厚度引起的光学效应,并简化了光线追踪计算。接近平行于主轴的会聚透镜的光线会偏离,因此它们会穿过主轴上的一个公共点,即主焦点。透镜有两个主焦点,一个用于从左侧入射的光,我们用 表示,第二个 用于从右侧入射的光。对于薄透镜,主焦点与透镜的光学中心等距,如图 15-9 所示。折射率被空气 包围的薄透镜的焦距 由透镜制造商方程给出
where and are the radii of curvature of the two surfaces. Here, is taken to be positive if the first surface is convex, and negative if the surface is concave. The signs are reversed for the back surface of the lens: is positive if the surface is concave, and negative if it is convex. The line is called as principal optical axis, which passes through the center of curvature of each surface. The front and rear (or back) focal planes are defined as the planes which
其中 是两个曲面的曲率半径。在这里, 如果第一个表面是凸的,则认为是正的,如果表面是凹的,则认为为负。镜片背面的符号相反: 如果表面是凹的,则为正,如果表面是凸的,则为负。这条线 被称为主光轴,它穿过每个表面的曲率中心。前后(或后)焦平面定义为
Fig. 15-9 The foci of thin lenses: (a) double-convex; (b) double-concave
图15-9 薄透镜的焦点:(a)双凸;(b) 双凹
are perpendicular to the principal optical axis and pass through the front and rear focal points, The line through the center of thin lens in Fig. 15-10 is defined as the secondary optical axis.
垂直于主光轴并穿过前后焦点,图15-10中穿过薄透镜中心的线 定义为次光轴。
Fig. 15-10 Formation of a real image by a converging lens
图15-10 聚焦透镜形成真实图像
We shall also consider only light rays that make small angles with the central axis. That is the paraxial approximation. Take a converging lens as an example. As shown in Fig. 15-10, an object outside the focus is a distance from the converging lens and an image is formed at a distance from the lens, which in this case is real and inverted. The image can be located by tracing of three standard rays:
我们还将只考虑与中轴成小角度的光线。这就是近轴近似。以会聚透镜为例。如图15-10所示,焦点 外的物体 与会聚镜头有一段距离 在距离镜头一定距离处形成图像 ,在这种情况下,图像是真实的和倒置的。可以通过追踪三条标准光线来定位图像:
(1) A ray from parallel to the principal axis is deviated so that it passes through the principal focus of a converging lens (or diverge as though it came from the principal focus of a diverging lens).
(1) 平行 于主轴的光线偏离,使其穿过会聚透镜的主焦点(或发散,就好像它来自发散透镜的主焦点一样)。
(2) The ray from which approaches the lens along the line through the principal focus is deviated so that it leaves the lens parallel to the principal axis.
(2)沿线通过主焦点 接近透镜的光线 偏离,使透镜平行于主轴。
(3) A ray through the optical center of lens passes through the lens undeviated.
For a thin lens, in the paraxial approximation, the object and image distances are lated by the equation
对于薄透镜,在近轴近似中,物体 和图像 距离 由方程表示
which is called as the thin-lens equation, and can be used with both converging and diverging lenses. The sign convention of object distance, image distance and focal distance used in this text is listed in Table 15-3.
Table 15-3 Sign Convention for Thin Lenses
表15-3 薄透镜的标志约定
Quantity 数量 Positive when  Negative when 阴性时
Object location
Object is in front of lens
Object is in back of lens
Image location  图像位置 Image is in back of lens
Image is in front of lens
Image height  图像高度 Image is upright 图像是直立的 Image is inverted 图像反转
Focal length  焦距 Converging lens 会聚透镜 Diverging lens 发散镜头
The lateral magnification is
which is the same as Eq. (15-10) for magnification by a mirror.
Fig. 15-11 Formation of a virtual image by a converging lens
图15-11 通过会聚透镜形成虚拟图像
When an object is inside the principal focus of a converging lens, the image formed is erect, virtual, and enlarged in Fig. 15-11. In this case, the image distance is negative. When an object is placed in front of a system of lenses such as two lenses whose principal axes coincide, the positive of the final image can be calculated by repeated use of the lens equation. The object for the second lens is the image formed by the first. If there is a third lens, the image formed by the second lens acts as its object. The final image formed by a very complicated optical system can be located by successive application of the lens (or mirror) equation. Great care must be taken to assign the proper sign to each distance.
当物体位于会聚透镜的主焦点内时,形成的图像是直立的、虚拟的和放大的,如图 15-11 所示。在这种情况下,图像距离为负。当一个物体被放置在镜头系统(例如主轴重合的两个镜头)的前面时,可以通过重复使用镜头方程来计算最终图像的正值。第二个镜头的对象是第一个镜头形成的图像。如果有第三个镜头,则由第二个镜头形成的图像充当其对象。由非常复杂的光学系统形成的最终图像可以通过连续应用透镜(或反射镜)方程来定位。必须非常小心地为每个距离分配适当的标志。
Example 15-3 An object is placed from a thin diverging lens of focal length . Find the image distance and the lateral magnification.
例 15-3 将物体放置 在焦距