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利用Mathematica设计天文光学系统:理论与方法
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《几何光学:利用Mathematica®设计天文光学系统理论与实践》这本书属于几何光学理论书籍的第三类,它不着重于基础理论的深入探讨,而是侧重于将数学背景与实际应用相结合,特别针对在天文学中广泛使用的光学系统设计。该书的独特之处在于,它采用了一种基于费马原理和称为“准像路径”(stigmatic paths)的非传统方法来处理三阶像差理论,这种方法使得读者能够更为直观地理解和操作设计光学组合所需的公式,避免了拉格朗日和哈密顿形式的复杂性以及塞德尔关系。 书中所用的软件Mathematica®是一个强大的计算工具,它在求解复杂的光学问题时发挥着关键作用。作者们通过实例展示,如何运用这种方法简化三阶像差分析,从而帮助读者设计出性能优良的天文望远镜和相机。章节10-12则深入讨论了拉格朗日和哈密顿形式的光学,尽管这些理论在光学、力学和偏微分方程理论上具有不可替代的价值,但本书更注重实际应用中的实用性和直观性。 这本书的目标读者是对天文光学系统设计感兴趣的工程师、科学家或学生,他们希望掌握有效利用Mathematica进行系统设计的方法,而无需过多依赖于理论推导。通过阅读本书,读者不仅能学习到几何光学的基础知识,还能提升解决实际光学问题的能力,这对于现代天文学和光学工程的发展具有重要意义。
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4 1 Fermat’s Principle and General Considerations …
Fig. 1.1 The refraction law
n
i
i'
t
t’
x
Fig. 1.2 Curve and broken
paths
M
M’
xx’
y
0
γ
γ
1
γ
λ = N
1 −
N
2
N
2
sin
2
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
0
the unique ray between
the points x ∈ M and x
∈ M
. The above two remarks and Fermat’s principle
imply that γ
0
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
0
) ≤ OPL(γ). (1.6)
Then, γ
0
is a ray if and only if (1.6) is satisfied for any broken line γ
1
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
1
.To
the contrary, suppose that
OPL(γ
0
) ≤ OPL(γ
1
),
for any broken line γ
1
. 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
1
= xyx
verifies the
condition
OPL(γ
0
) ≤ OPL(γ
1
) ≤< OPL(γ).
![](https://csdnimg.cn/release/download_crawler_static/11109262/bg11.jpg)
1.1 An Introduction to Fermat’s Principle 5
4. Let S be a surface separating two regions C
1
and C
2
with constant refractive
indices N
1
and N
2
, respectively. We denote the family of all broken lines starting
from a point x
1
∈ C
1
and ending at the point x
2
∈ C
2
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
x
∈ 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
) =
x
x
Nt · dx, (1.7)
is a function of x and x
. Moreover, the following relations hold:
∇
x
V (x, x
) =−N(x)t(x), ∇
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
![](https://csdnimg.cn/release/download_crawler_static/11109262/bg12.jpg)
6 1 Fermat’s Principle and General Considerations …
Fig. 1.3 Arayγ from the
object plane to the image
plane
x’
y’
O’
’
O
x
y
a
t
γ
π
π
from (x, y) is determined by providing the unit vector t along γ, i.e., two director
cosines α and β of γ (see Figure 1.3). It should be noted that the quantities x, y, t,
although natural, are not the most convenient variables to use. In fact, it is evident
that some rays coming from (x, y) ∈ π do not reaches the optical system S, due to
either the finite dimensions of the optical components or the presence of a stop along
the optical path.
Let us consider the set of the rays arising from the point O ∈ a. In Figure 1.4
the rim of the first surface of the optical path limits the cone C of rays from O.It
is then natural to identify a ray γ by providing the coordinates (x, y) of the point
of the object in π and the coordinates (x
e
, y
e
) of the point of intersection of γ with
the tangent plane π
e
at the vertex of the first surface of S. In Figure 1.5, the cone C
is determined by a stop in the plane π
e
, and we can again identify (x
e
, y
e
) with the
coordinates of the intersection point of a ray with π
e
. However, in Figure 1.6 the stop
is located after the optical system. In this case, owing to Fermat’s principle, a ray
Fig. 1.4 Stop on the first
surface
O
a
y
y’
O’
t
e
γ
π
x
e
, y ()
e
Fig. 1.5 Stop before the first
surface
O
a
y
y’
O’
t
e
x
e
, y ()
e
stop
γ
π
![](https://csdnimg.cn/release/download_crawler_static/11109262/bg13.jpg)
1.2 Forming an Image 7
Fig. 1.6 Stop behind the last
surface
O
a
y
y’
O’
t
e
()x,y
ee
stop
π
γ
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
e
,in
front of the object plane π, and determine the coordinates (x
e
, y
e
) of the point of
intersection of γ with π
e
of any ray which succeeds in crossing S. In Chapter 2 we
discuss the criterion for defining a suitable reference plane π
e
. 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
e
in the cases of Figures 1.4, 1.5, whereas it
gives the correct position of the plane π
e
in the case of Figure 1.6. In conclusion,
from now on, a ray will be identified by the variables x, y, x
e
, y
e
.
Let π
be another plane orthogonal to the optical axis a. Two points x ∈ π and
x
∈ 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
y
z
that have their origins O, O
in
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
e
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
e
, y
e
) or the polar coordinates (r, ϕ) of the
![](https://csdnimg.cn/release/download_crawler_static/11109262/bg14.jpg)
8 1 Fermat’s Principle and General Considerations …
Fig. 1.7 Meridional rays
x’
y’
O’
’
O
x
y
meridionalplane
meridionalray
π
π
Fig. 1.8 Sagittal rays
x’
y’
O’
’
O
x
y
sagittal lanep
sagittalray
π
π
point at which r intersects with π
e
.Let(x
, y
) be the point of the image plane π
reached by r. Using these notations, we can say that the system S is ideal if
x
= Mx,
y
= My,
(1.9)
where M, which is called the magnification of S with respect to the pair of conjugate
planes π and π
, is independent of (x, y), (x
e
, y
e
), and the wavelength of light λ (i.e.,
of the object point, the ray, and the wavelength).
In other words, an optical system is ideal if the the following properties hold:
• The image of the object is planar and does not exhibits distortion;
• Any point of the object and its corresponding point on the image are stigmatic.
• The above properties are independent of the wavelength of light λ.
In the following chapter it is shown that, for a given wavelength λ and object plane
π, any centered optical system is almost ideal; in other words, there is a unique
image plane π
—termed the Gaussian image plane—for which the first two of the
above properties are true, at least for points on the object and image belonging to
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