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Problem (1) gives 问题 (1) 给出
Therefore 因此
This is the wavelength of violet. The first diffraction maximum of violet light with wavelength will always coincide with the first minimum of red light with wavelength , no matter what the slit width is. If the slit is relatively narrow, the overlapping of the angle will be relatively large, and conversely. Therefore, each fringe is a rainbow of color except for the central maximum when white light falls on a slit.
这是紫罗兰的波长。无论狭缝宽度如何,波长 的紫光的第一个衍射最大值总是与波长 的红光的第一个最小值一致。如果狭缝相对较窄,则角度 的重叠将相对较大,反之亦然。因此,除了白光落在狭缝上的中心最大值外,每个条纹都是彩虹色。

15.11 Diffraction Grating
15.11 衍射光栅

A logical extension of Young's double-slit interference experiment is to increase the number of slits from two to a large number of . An optical component consisting of a large number of parallel, closely spaced slits- for example, as many as is not uncommon is called a diffraction grating. A diffraction grating can be used to determine the wavelength of light with high precision. Diffraction gratings are usually made by ruling equally spaced parallel grooves on a polished glass plate using a diamond-tipped cutting tool. The grooves are effectively opaque and they scatter the light and the space between the grooves behaves as a slit. Thus, the action of a diffraction grating can be described in terms of a regular array of parallel slits. The width of a slit is and the width of a groove is is called grating constant as illustrated in Fig. 15-36. Usually has orders of , meaning that there are grooves per centimeter. Gratings are widely used to measure wavelengths and to study the structure and intensity of spectral lines. Intensity patterns of bright and dark fringes can be seen on the viewing screen when monochromatic light passes through a single or double slit. The fringe patterns also result when light falls on a grating. Fig. 15-37 shows a grating with total number of slits . At any point on the screen, the available light intensity from each slit, considered separately, is given by the diffraction pattern of that slit. The diffraction patterns for each separate slit coincide with each other because parallel rays in Fraunhofer diffraction are focused on the same point of the lens' focus plane. The diffraction rays will interference with each other since they are coherent. The diffraction patterns for equals and 20 are shown respectively in Fig. 15-38. Two important changes occur when the number of slits increases:
Young 双缝干涉实验的一个逻辑扩展是将狭缝的数量从两个增加到大量 。由大量平行、紧密间隔的狭缝组成的光学元件(例如,数量 不少见)称为衍射光栅。衍射光栅可用于高精度地确定光的波长。衍射光栅通常是通过使用金刚石尖端切割工具在抛光玻璃板上划出等距的平行凹槽制成的。凹槽实际上是不透明的,它们会散射光线,凹槽之间的空间表现为狭缝。因此,衍射光栅的作用可以用平行狭缝的规则阵列来描述。狭缝的宽度 和凹槽 的宽度称为光栅常数,如图 15-36 所示。通常 有 的阶数, 表示每厘米有 凹槽。光栅广泛用于测量波长和研究光谱线的结构和强度。当单色光通过单缝或双缝时,可以在观看屏幕上看到明色和暗色条纹的强度模式。当光线落在光栅上时,也会产生条纹图案。图 15-37 显示了具有狭缝总数的光栅 。在屏幕上的任何一点,每个狭缝的可用光强度(单独考虑)由该狭缝的衍射图给出。每个独立狭缝的衍射图相互重合,因为弗劳恩霍夫衍射中的平行光线聚焦在透镜焦平面的同一点上。衍射射线会相互干涉,因为它们是相干的。 值和20的衍射图分别如图15-38所示。当狭缝数量增加时,会发生两个重要变化:
(1) The bright fringes become narrower and brighter;
(1)明亮的条纹变窄,变亮;
(2) Faint secondary maxima appear between fringes.
(2)条纹之间出现微弱的次级极大值。
It indicates that the diffraction pattern of a grating is total result of interference of the lights from slits and diffraction of a single slit. The more slits, the more principal maxima and the weaker the secondary maxima become. As increase, perhaps to for a useful grating, the bright fringes become very sharp and bright indeed while the secondary maxima become so
它表明光栅的衍射图是狭缝光的干涉和单个狭缝的衍射的总结果。狭缝越多,主最大值越多,次级最大值越弱。随着 增加,也许对于 有用的光栅,明亮的条纹变得非常清晰和明亮,而次级最大值则变得如此

reduced in intensity as to be negligible in their effects. We will ignore the secondary maxim in what follows and discuss the locations of the bright fringes of a diffraction grating. The optical path difference between rays from adjacent slits shown in Fig. 15-38 is
强度降低,其影响可以忽略不计。我们将在下文中忽略次要格言,并讨论衍射光栅的明亮条纹的位置。图15-38所示的相邻狭缝光线之间的光程差为
Fig. 15-36 Grating constant
图15-36 光栅常数
Fig. 15-38 Diffraction pattern for (from up to bottom)
图15-38 衍射图 (从上到下)
Fig. 15-37 Arrangement for a grating of
图15-37 光栅 的布置
where is diffraction angle. Constructive interference creates bright fringes. These sharp bright fringes which are sometimes called principal maxima will occur when
其中 是衍射角。相长干涉会产生明亮的条纹。这些尖锐的明亮条纹有时被称为主极大值,当
Where is the wavelength of incident light, is the order of the fringe (principal maximum) and corresponding to the central fringe (or central principal maximum). Eq. (15-35) is called grating equation which gives the condition necessary to obtain constructive interference for a diffraction grating in the case of normal incidence.
其中 是入射光的波长, 是条纹的阶数(主极大值), 对应于中心条纹(或中心主极大值)。方程(15-35)称为光栅方程,它给出了在法向入射的情况下获得衍射光栅的相长干涉的必要条件。
Missing orders occur for a diffraction grating when an interference maximum coincides with a diffraction minimum of a single slit. For example, if both the conditions and are satisfied for given , the th principal maximum of the grating diffraction is coincident with the th minimum of the gle-slit diffraction. As a result, there is no the th prin-
当干涉最大值与单个狭缝的衍射最小值重合时,衍射光栅会出现缺序。例如,如果两个条件 都满足, 则光栅衍射的 主最大值与 格栅狭缝衍射的 最小值重合。因此,没有 th prin-
cipal maximum on the viewing screen, that is such bright fringe disappears.
在观看屏幕上的最大值,即如此明亮的条纹消失。
Eq. (15-35) can be used to study the dependence of the diffraction angle on the wavelength . When the grating constant keeps a constant, the diffraction angle of the th bright fringe is determined only by the wavelength . The longer the wavelength is, the lar
式(15-35)可用于研究衍射角 对波长 的依赖性。当光栅常数 保持恒定时, 第个明亮条纹的衍射角 仅由波长 决定。波长越长,波长 越长

ger the angle is. When we use white light as incident light, except for the central fringe keeps white, all the higher-orders maxima disperse white light into their rainbow of colors on the screen. The whole of all the same order fringe of the incident light emitted by a polychromatic source is called grating spectrum, the fringe on the grating spectrum is called spectrum line. The grating spectrum formed by one material is unique, that is, a kind of material has a characteristic spectrum. We can determine components of material by its grating spectrum.
蒙古角 是。当我们使用白光作为入射光时,除了中央边缘保持白色外,所有高阶最大值都会将白光分散到屏幕上的彩虹色中。由多色光源发射的入射光的所有相同阶的条纹称为光栅光谱,光栅光谱上的条纹称为光谱线。一种材料形成的光栅光谱是唯一的,即一种材料具有特征光谱。我们可以通过光栅光谱来确定材料的成分。
Example 15-13 A beam of white light ( ) falls normally on a grating, as shown in Fig. 1539. The grating constant is , the focal length is . Find the distances from the order spectrum line of violet light ( ) and the 2nd order spectrum line of red light ( ) to the centre on the screen, respectively.
实施例15-13 一束白光( )正常落在光栅上,如图1539所示。光栅常数为 ,焦距为 。分别求紫光的 阶谱线 ( ) 和红光的二阶谱线 ( ) 到屏幕中心的 距离。
Fig. 15-39 for Example 15-13
图 15-39 示例 15-13
Solution Assuming the diffraction angles of the 3 rd order spectrum line of the violet light and the 2 nd order spectrum line of the red light are and , respectively. The distance from them to the centre are and , respectively. From the grating Eq. (15-35), we have
解 假设紫光的 3 阶光谱线和红光的 2 阶光谱线的衍射角分别为 。从它们到中心的 距离分别是 。从光栅方程(15-35)中,我们有
From the geometry shown in Fig. 15-39, we have
从图 15-39 所示的几何形状中,我们有
We can see that is smaller than . It indicates that the 3 rd order spectrum line of the violet light is more near the central maximum line than the 2 nd order spectrum line of the red light and the two spectrums overlap with each other. Please note that when is very small (e. g. ), the relationship that is used in discussing of single-slit diffraction may not suitable for discussing of diffraction grating.
我们可以看到它 小于 。它表明紫光的3阶光谱线比红光的2阶光谱线更靠近中心最大线,并且两个光谱相互重叠。请注意,当 非常小时(例如 ),用于讨论单缝衍射的关系 可能不适合讨论衍射光栅。
Example 15-14 Light of wavelength is incident obliquely on a grating at an angle as shown in Fig. 15-40. The diffraction grating has 5000 lines per centimeter.
实施例15-14 波长 的光以一定角度 斜入射在光栅上,如图15-40所示。衍射光栅每厘米有 5000 条线。
Fig. 15-40 For Example
图 15-40 举例
(1) Find the diffraction angles for bright fringes corresponding to and , respectively.
(1)求亮条纹的衍射角分别对应于
(2) Find the highest order of maximum that can be seen. Compare it with the one in the case of normal incidence.
(2)找到可以看到的最大值的最高阶数。将其与正常发生率的情况进行比较。
Solution (1) As light is oblique incidence, the optical path difference between rays from adjacent slits in Fig. 15-40 is
解决方案(1)由于光是斜入射的,因此图15-40中相邻狭缝的光线之间的光程差为
The principal maxima will occur when
Eq. (15-36) is called grating equation for oblique incidence. In this problem, the grating con.
方程(15-36)称为斜入射光栅方程。在这个问题上,光栅骗局。

stant 斯坦特
From Eq. (15-36), the diffraction angle for fringe of is
由式(15-36)可知,条纹 的衍射角为
It indicates the 0th order bright fringe located below the centre of the screen.
它表示位于屏幕中心下方的 0 阶亮条纹。
The diffraction angles corresponding to fringes of and are respectively
对应于 和 条纹的衍射角分别为
and
These tell us that the distribution of bright fringes is not symmetrical.
这些告诉我们,明亮条纹的分布是不对称的。
(2) The highest order of maximum corresponds with the maximum of , that is . From Eq. (15-36), we have
(2) 最大值的最高阶数对应 于 的最大值 ,即 。从式(15-36)中,我们有
This tells us that the highest order of maximum which can be seen is the fourth-order fringe when oblique incidence occurs.
这告诉我们,当斜入射发生时,可以看到的最高阶最大值是四阶条纹。
Assuming the highest order of the spectral line is in the case of normal incidence, according to the grating equation for normal incidence
假设谱线的最高阶是在 正常入射的情况下,根据正常入射的光栅方程
we have 我们有
which means that the highest order of maximum which can be seen is the third-order fringe when normal incidence occurs,
这意味着当正常入射发生时,可以看到的最大值的最高阶是三阶条纹,
Therefore, we can observe higher order of bright fringes in the case of oblique incidence. The highest order of spectral line is only decided by the incident angle when the wavelength and the grating constant are unchanged,
因此,在斜入射的情况下,我们可以观察到更高阶的明亮条纹。当波长 和光栅常数 不变时,光谱线的最高阶仅由入射角 决定,

15.12 Resolving Power of Optical Instruments
15.12 光学仪器的分辨能力

Diffraction comes from the wave nature of light and is determined by the finite aperture of the optical elements. An important property of any optical instrument, such as a camera or telescope, is its resolving power. Resolving power is the ability to distinguish between two closely spaced objects. We have known that light passing a small opening or a boundary of circular lens is bended so that the images are fuzzy. Diffraction fringes near the edges of in
衍射来自光的波动性质,由光学元件的有限孔径决定。任何光学仪器(例如相机或 望远镜)的一个重要特性是其分辨能力。分辨能力是区分两个紧密间隔的物体的能力。我们已经知道,通过圆形透镜的小开口或边界 的光会弯曲,因此图像是模糊的。边缘 附近的衍射条纹

ges usually make it difficult to determine the exact shape of the source. It limits the resolving power of an optical instrument because the diffraction pattern has certain intensity distribution. Fig. 15-41 shows a circular diffraction pattern formed by a small circular aperture. Note that the large central maximum (bright spot), called Airy disk, is surrounded by alternating bright and dark rings. This diffraction is of extreme importance in an optical instrument because it sets the ultimate limit on the possible magnification.
GES通常很难确定源的确切形状。它限制了光学仪器的分辨能力,因为衍射图具有一定的强度分布。图15-41显示了由小圆孔形成的圆衍射图。请注意,大的中心最大值(亮点),称为艾里盘,被交替的亮环和暗环包围。这种衍射在光学仪器中极为重要,因为它为可能的放大倍率设定了最终限制。
Fig. 15-41 Diffraction pattern of a circular aperture
图15-41 圆形孔径的衍射图
Fig. 15-42 shows that light from two sources and pass through a small circular opening in an opaque barrier. In Fig. 15-42 (a), the images of the sources and are distinguished as separate images. In this situation, the sources are said to be resolved. If they are brought closer together as shown in Fig. 15-42 (b), however, their images overlap resulting in a confused image. When the sources are so close together (or the opening is so small) that the separate images can no longer be distinguished, the sources are said to be unresolved. The resolving power of an optical instrument is a measurement of its ability to produce well-defined separate images.
图15-42显示了来自两个光源 的光,并通过 不透明屏障中的一个小圆形开口。在图15-42(a)中,图像源 图像被区分为单独的图像。在这种情况下,据说来源已解决。但是,如果将它们拉得更近,如图15-42(b)所示,则它们的图像会重叠,从而产生混淆的图像。当源距离如此之近(或开口如此之小)以至于无法再区分单独的图像时,则称源未解析。光学仪器的分辨能力是衡量其产生清晰定义的单独图像的能力的指标。
(a)
(b)
Fig. 15-42 (a) The images of the sources and are easily distinguished;
图15-42 (a)来源 的图像, 易于区分;
(b) As the sources are brought closer together, the images overlap, resulting in a confused image.
(b) 当来源靠得更近时,图像重叠,导致图像混淆。
It can be proved that the first minimum of the diffraction pattern of a circular aperture with a diameter (Fig. 15-43) is given by
可以证明,直径为圆形孔径 的衍射图的第一个最小值(图15-43)由下式给出
Fig. 15-43 Ariy disk
图15-43 Ariy圆盘
From Eq. (15-37) propor se that the size of the central maximum is directly portional to . That is, the central maximum is more spread out for longer wavelength and smaller aperture. No matter how perfectly a lens is constructed, the image of a point source of light will not be focused at a point. What is the condition of two images just re-
从式(15-37) 可以看出,中心最大值的大小与 成正比。也就是说,对于更长的波长和更小的孔径,中心最大值更加分散。无论镜头构造得多么完美,点光源的图像都不会聚焦在一个点上。两张图片的状况是什么,只是重新

solved? The accepted criterion for resolution is the Rayleigh criterion, first proposed by Rayleigh (1842-1919): two images are just resolved when the center of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other. It tells us that two images are just resolved when the center of central maximum of one pattern coincides with the first dark fringe of the other. According to Rayleigh criterion and Eq. (15-37), two point objects separated by an angle are just resolved when
解决?公认的分辨率标准是瑞利准则,由 瑞利(1842-1919)首次提出:当一个图像的衍射图的中心正好位于另一个图像的衍射图的第一个最小值上时,两个图像刚刚被解析。它告诉我们,当一个模式的中心最大值中心与另一个模式的第一个暗条纹重合时,两个图像就会被解析。根据瑞利准则和方程(15-37),
where