利用Mathematica设计天文光学系统：理论与方法

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4 1 Fermat’s Principle and General Considerations …

Fig. 1.1 The refraction law

t’

Fig. 1.2 Curve and broken

paths

M’

xx’

λ = N



1 −

2

sin

i − N cos i. (1.5)

Relations (1.3)–(1.5), which express the refraction law, will be considered again

in Chapter 11.

3. Let S be a surface separating two homogeneous media M and M



with refractive

indices N and N



, respectively, (see Figure 1.2) and γ

the unique ray between

the points x ∈ M and x



∈ M



. The above two remarks and Fermat’s principle

imply that γ

is a broken line whose optical path length is smaller than or equal

to the optical path length of any curve γ with the same end points, i.e.

OPL(γ

) ≤ OPL(γ). (1.6)

Then, γ

is a ray if and only if (1.6) is satisﬁed for any broken line γ

between

the same points x and x



In fact, if (1.6) is veriﬁed by any curve γ it is veriﬁed by any broken line γ

.To

the contrary, suppose that

OPL(γ

) ≤ OPL(γ

for any broken line γ

. Then, let γ be any curve between x and x



and denote by

y the intersection point of γ with S. Then, the broken line γ

= xyx



veriﬁes the

condition

OPL(γ

) ≤ OPL(γ

) ≤< OPL(γ).

1.1 An Introduction to Fermat’s Principle 5

4. Let S be a surface separating two regions C

and C

with constant refractive

indices N

and N

, respectively. We denote the family of all broken lines starting

from a point x

∈ C

and ending at the point x

∈ C

by . If the optical path

length OPL(γ), with γ ∈ , is independent of γ, then all the broken lines of 

are rays. In fact, since OPL(γ) has the same value on any γ ∈ , it is stationary

along any path.

Under these conditions, the initial and ﬁnal points of the rays belonging to  are

said to be stigmatic with respect to S.

5. Let I (x) be a spatial region containing the point x. Suppose that for any point



∈ I (x) one and only one ray γ exists between x and x



. Then, denoting the

unit tangent vector to γ by t, the optical path length along γ

V (x, x



) =





Nt · dx, (1.7)

is a function of x and x



. Moreover, the following relations hold:

∇

V (x, x



) =−N(x)t(x), ∇



V (x, x



) = N(x



)t(x



), (1.8)

Formulae (1.8), which derive from the well-known Leibnitz’s rule, will be proven

in Chapter 11.

Remark 1.1 We note that the reﬂection law, according to which the incident angle

i and the reﬂection angle i



satisfy the condition i



=−i, can be obtained by the

refraction law (1.4) by putting into it N =−N



= 1.

In Chapters 2-9 we show many applications of the above consequences of Fermat’s

principle.

1.2 Forming an Image

Throughout this book, we consider centered optical systems S consisting of surfaces

with a common axis a of revolution that separate homogeneous optical media with

different (but constant for each medium) refractive indices. The axis a is called the

optical axis of S. We also suppose that the object that S must image is contained in

a plane π, which is orthogonal to a and is called the object plane.

All the considerations discussed in the following sections of this chapter refer to

a given optical system S with the above characteristics.

To analyze the behavior of the rays originating from a point of the object and

crossing the optical system S, we can use the refraction and reﬂection laws. This

procedure, which is called ray tracing, will be analyzed in detail in Section 1.6.Itis

evident that it requires that the mathematical quantities that deﬁne a ray are assigned.

Now, if we introduce Cartesian axes Oxy in the object plane π, then a ray γ coming

1.2 Forming an Image 7

Fig. 1.6 Stop behind the last

surface

y’

O’

()x,y

stop

from O is still determined by providing the coordinates of its point of intersection

with the plane of the stop, but we cannot explicitly write the equation of the ray

by providing these coordinates. In this case, we must choose a suitable plane π

,in

front of the object plane π, and determine the coordinates (x

, y

) of the point of

intersection of γ with π

of any ray which succeeds in crossing S. In Chapter 2 we

discuss the criterion for deﬁning a suitable reference plane π

. This analysis, which

lead us to the deﬁnitions of entrance pupil and exit pupil in particular, shows that

the reference plane is just given by π

in the cases of Figures 1.4, 1.5, whereas it

gives the correct position of the plane π

in the case of Figure 1.6. In conclusion,

from now on, a ray will be identiﬁed by the variables x, y, x

, y

Let π



be another plane orthogonal to the optical axis a. Two points x ∈ π and



∈ π



are said to be stigmatic for S if all of the rays from x reach the point x



after

crossing the system S. The system S is said to be an ideal system or an aberration-free

system for the pair of the object plane π and the plane π



if the following conditions

are satisﬁed:

1. There is a plane π



such that points x ∈ π exhibit a one-to-one correspondence

with points in π



, independent on wavelength of the light. In other words, each

point x



∈ π



corresponds to one and only one point x ∈ π.

2. The above correspondence preserves the form of the object.

In this case, the plane π



is the image plane and π and π



are said to be conjugate

planes.

Consider two Cartesian frames Oxyzand Ox



that have their origins O, O



π, π



, respectively, and that have the axes Oz, Oz



coinciding with the optical axis a

of S. The coordinate plane Oyzis called the meridional or tangential plane whereas

Oxz is said to be the sagittal plane. Due to the refraction law, the rays which are

present in the plane Oyzbefore meeting S remain in this plane after crossing S.They

are called meridional or tangential rays whereas the rays that lie in the plane Oxz

are said to be sagittal rays (see Figures 1.7 and 1.8). All the other rays are called

skew rays.

The region containing the object, which by convention is always supposed posi-

tioned on left-hand side of S,istheobject space whereas the region containing the

image formed by S is the image space.

If π

is the above mentioned auxiliary plane in the object space, then any ray

r coming from the point (x, y) in the object plane π is completely determined by

providing the Cartesian coordinates (x

, y

) or the polar coordinates (r, ϕ) of the

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