计算光学：MATLAB与Maple实践教程

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"本书《基于MATLAB, Maple, MathCad的计算光学》是学习光学与编程结合的实用教程，特别适合于对MATLAB, MATHCAD等软件感兴趣的读者。作者K.D.Moller通过308个插图和配套CD-ROM中的Mathcad, Matlab, Mathematica及Maple软件实例，深入浅出地讲解了光学理论及其计算方法。" 本书详细介绍了光学的基本概念，包括几何光学，这是光学的基础部分，主要研究光的直线传播、反射、折射等现象。在数据处理方面，书中利用现代计算工具如MATLAB、Maple和Mathematica进行数据分析和模拟，使读者能够更直观地理解和应用光学原理。 MATLAB是一款强大的数值计算和可视化软件，常用于科学计算和工程问题的解决。在光学领域，MATLAB可以用来模拟光的传播路径、计算衍射图案、设计光学系统等。书中通过MATLAB实例，帮助读者掌握如何利用该软件进行光学计算。 Maple是一款通用的数学软件，它提供了丰富的数学函数库和图形界面，使得复杂的数学运算和符号运算变得简单。在光学学习中，Maple可以帮助解决涉及复数、矩阵和微积分的问题，例如计算光的干涉和衍射。 MathCad则是专为工程和科学计算设计的一款软件，它的特点是易于使用且具有良好的公式排版功能。在光学学习中，MathCad可用于构建光学模型，进行光学参数的计算，以及创建清晰的计算报告。 Mathematica同样是一款强大的数学软件，其强大的符号计算能力和图像生成功能在光学理论分析中非常有用。例如，它可以用来求解波动光学中的偏振、傅里叶变换以及光场的复数表示等问题。此书不仅涵盖了光学的基本理论，还强调了计算机辅助计算的重要性，旨在培养读者的编程能力和将理论应用于实际问题的能力。书中的例子涵盖了光学的主要分支，包括几何光学、波动光学和量子光学等方面，使得读者能够全面了解和掌握光学知识，并通过实际操作提高计算技能。此外，附带的CD-ROM包含了所有示例的源代码，方便读者直接运行和学习，这无疑增加了本书的实践性和互动性。这本书是学习光学并结合现代计算工具的理想教材，无论是对初学者还是有一定基础的读者，都能从中受益匪浅。

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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