利用Mathcad、Matlab、Mathematica和Maple解决光学问题的计算学习法

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"《光学：通过计算学习实例——使用MathCAD、Matlab、Mathematica与Maple》是一本由Karl Dieter Möller编著的第二版教材，专注于在光学领域内运用计算机辅助学习的方法。本书旨在帮助读者通过实际问题解决来深入理解几何光学原理，利用Mathcad、Matlab、Mathematica和Maple等高级数学软件工具进行计算和建模。该书特别关注于如何利用这些软件进行数据处理，通过丰富的实例展示如何将理论知识转化为实践操作。书中涵盖了123个精心设计的练习，涵盖了诸如光线传播、反射、折射、干涉和衍射等基本光学概念，以及现代光学技术中的应用。每一部分都强调了动手操作的重要性，以便读者能够熟练掌握这些软件在光学分析中的应用技巧。《光学：学习通过计算》不仅是一本教科书，也是一份实用的手册，适合大学物理专业的学生、工程师和技术人员使用，他们可以借此提升自己的数学技能，掌握数值模拟和可视化工具。此外，对于那些希望了解如何用科学计算语言来解决光学问题的研究人员，这本书也是一个宝贵的参考资源。作者Karl Dieter Möller是新泽西理工学院物理学系的教授，他的专业背景和教学经验使得本书内容既严谨又易于理解。书中还包含了详细的索引和参考文献，方便读者进一步探索相关领域的深度知识。出版商Springer Science+Business Media提供了版权保护，确保了该书的高质量印刷，并提供了包含Mathcad软件的配套光盘，便于读者在学习过程中实时操作和实践。无论是作为课堂教材还是个人学习资料，这本书都是学习和研究光学计算的优秀选择，强调了理论与实践的结合，有助于读者在信息技术驱动的科学探索中不断进步。"

CHAPTER

Geometrical

Optics

1.1 INTRODUCTION

Geometrical optics uses light rays to describe image formation by spherical

surfaces, lenses, mirrors, and optical instruments. Let us consider the real image

of a real object, produced by a positive thin lens. Cones of light are assumed to

diverge from each object point to the lens.There the cones of light are transformed

into converging beams traveling to the corresponding real image points. We

develop a very simple method for a geometrical construction of the image, using

just two rays among the object, the image, and the lens. We decompose the object

into object points and draw a line from each object point through the center of

the lens. A formula is developed to give the distance of the image point, when

the distance of the object point and the focal length of the lens are known. We

assume that the line from object to image point makes only small angles with the

axis of the system. This approximation is called the paraxial theory. Assuming

that the object and image points are in a medium with refractive index 1 and that

the lens has the focal length f, the simple mathematical formula

−x



(1.1)

gives the image position x

when the object position x

and the focal length are

known.

Formulas of this type can be developed for spherical surfaces, thin and thick

lenses, and spherical mirrors, and one may call this approach the thin lens model.

For the description of the imaging process, we use the following laws.

1. Light propagates in straight lines.

2. The law of refraction,

sin θ

 n

sin θ

. (1.2)

2 1. GEOMETRICAL OPTICS

The light travels through the medium of refractive index n

and makes the an-

gle θ

with the normal of the interface.After traversing the interface, the angle

changes to θ

, and the light travels in the medium with refractive index n

3. The law of reﬂection

 θ

. (1.3)

The law of reﬂection is the limiting case for the situation where both refraction

indices are the same and one has a reﬂecting surface. The laws of refraction

and reﬂection may be derived from Maxwell’s theory of electromagnetic

waves, but may also be derived from a “mechanical model” using Fermat’s

Principle.

The refractive index in a dielectric medium is deﬁned as n  c/v, where v

is the speed of light in the medium and c is the speed of light in a vacuum. The

speed of light is no longer the ratio of the unit length of the length standard over

the unit time of the time standard, but is now deﬁned as 2.99792458 × 10

m/s

for vacuum. For practical purposes one uses c  3 × 10

m/s, and assumes that

in air the speed v of light is the same as c. In dielectric materials, the speed v is

smaller than c and therefore, the refractive index is larger than 1.

Image formation by our eye also uses just one lens, but not a thin one of ﬁxed

focal length. The eye lens has a variable focal length and is capable of forming

images of objects at various distances without changing the distance between the

eye lens and the retina. Optical instruments, such as magniﬁers, microscopes,

and telescopes, when used with our eye for image formation, can be adjusted

in such a way that we can use a ﬁxed focal length of our eye. Image formation

by our eye has an additional feature. Our brain inverts the image arriving on the

retina, making us think that an inverted image is erect.

1.2 FERMAT’S PRINCIPLE AND THE LAW OF

REFRACTION

In the seventeenth century philosophers contemplated the idea that nature always

acts in an optimum fashion. Let us consider a medium made of different sections,

with each having a different index of refraction. Light will move through each

section with a different velocity and along a straight line. But since the sections

have different refractive indices, the light does not move along a straight line

from the point of incidence to the point of exit.

The mathematician Fermat formulated the calculation of the optimum path as

an integral over the optical path



nds. (1.4)

4 1. GEOMETRICAL OPTICS

medium in which its speed is v

, the angle with respect to the normal changes

from θ

to θ

Let us look at a popular example. A swimmer cries for help and a lifeguard

starts running to help him. He runs on the sand with v

, faster than he can swim

in the water with v

. To get to the swimmer in minimum time, he will not choose

the straight line between his starting point and the swimmer in the water. He will

run a much larger portion on the sand and then get into the water. Although the

total length (in meter’s) of this path is larger than the straight line, the total time

is smaller. The problem is reduced to what the angles θ

and θ

are at the normal

of the interface (Figure 1.1). We show that these two angles are determined by

the law of refraction, assuming that the velocities are known.

In Figure 1.1 the light from point P

travels to point P

and passes the point Q

at the boundary of the two media with indices n

and n

. The velocity for travel

from P

to Q is v

 c/n

. The velocity for travel from Q to P

is v

 c/n

From Eq. (1.4) and Figure 1.1, the optical path is

(y) + n

(y), (1.7)

where we have

(y) 



+ y

}

(y) 



{(x

− x

)

+ (y

− y)

} (1.8)

and with r

(y)  v

(y) and r

(y)  v

(y) we get for the total time T (y), to

travel from P

to P

T (y)  r

(y)/v

+ r

(y)/v

. (1.9)

Only for the special case that v

 v

, where the refractive indices are equal,

will the light travel along a straight line. For different velocities, the total travel

time through medium 1 and 2 will be a minimum. In FileFig 1.1 we show a graph

of T (y) and see the minimum for a speciﬁc value of y. In FileFig 1.2 we discuss

the case where light is traveling through three media. To determine the optimum

conditions we have to require that

dT (y)/dy  0. (1.10)

This may be done without a computer. We show it in FileFig 1.3 for two media.

Using the expression for r

(y) and r

(y) of Figure 1.1, we have to differentiate

(y) + n

(y), (1.11)

that is,

dT (y)/dy  d/dy{(c/v

)



+ y

+ (c/v

)



− x

)

+ (y

− y)

} (1.12)

and set it to zero. From FileFig 1.3 we get

y/(r

(y)v

) + (y − y

)/(r

(y)v

)  0. (1.13)

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