逢 甲 大 學
電機工程學系碩士班
碩士論文
Zemax 軟體模擬攝影鏡頭的設計
The Revolution and Design of Photographic Lens
with Zemax

指導教授：李企桓
研 究 生：Dang Xuan Du


中

華

民

國

一

百

零

五

年

六

月


The Revolution and Design of Photographic Lens with Zemax

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The Revolution and Design of Photographic Lens with Zemax

Acknowledgement
This thesis does not just reflect the works done at the Opto-electronic System
Design Laboratory-Feng Chia University including many project experiences, but it
also represents a personally enriching and unforgettable time in Taichung, Taiwan.
Firstly, I would like to thank to Feng Chia University for giving me scholarship
to complete my education.
My deepest appreciation goes to my advisor Professor 李企桓 (Chi-Hung Lee)
for accepting me as his student. He has provided me the valuable information and
suggestions for my research.
My gratitude goes to all the lecturers who have taught me and staffs in
Department of Electrical Engineering for their friendly support. Their useful
comments and suggestions improved the quality and contents of this research.
Thanks also go to all my lab-mate who are very kind and give me supports
enthusiastically. They were and are helping me to solve many problem in learning as
well as in other fields.
Especially, I want to thank to my parent, my relatives for their continuous and
unquestioning support of my study. It is more than the moral support and constant
encouragement.


Dang Xuan Du 杜光東

電機工程學系碩士班
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The Revolution and Design of Photographic Lens with Zemax

Abstract
Before the breakthrough of photography, pictures were rare and exclusive. All portrait
or landscape picture was created by the oblique way, the ingenious hands. But with the
invention of lens, a little piece of glass that would change the world. The lens started a
formidable revolution in our ability to explore our surroundings, increase our
knowledge, and gradually made it possible to alter our circumstances in a positive way.
Since the invention of lens, it has been undergone many revolutions that was in ordered
to satisfy the needs and requirements on its own time.
Each of new type of lens was invented to solve the drawbacks of the previous lens, then
improve the quality and performance. So, in chapter two, we are going to redesign some
of these important lenses by Zemax, analyze their properties, estimate their quality to
have a clearer understanding in the development progress of lens. Consequently, a series
of lens has been designed and shown up already.
In the next chapter, the discussion about related optic theories that useful for optical
design. The methods were used to do the works in this study also mentioned. By
combination between the own optical knowledge, design skills, applying new
technology and the powerful of software, the lens has approached a good quality of
performance.
Tessar lens and Cooke Triplet lens were the two outstanding lens and were widely used

in many applications, so a research and design new versions of these lens will be the
main tasks in this study.
Keywords: Photographic lens, Cooke Triplet lens, Tessar lens, history of lens

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CONTENTS
Chapter 1: INTRODUCTION ................................................................................. 1
1.1

Review of photographic lens ............................................................................ 1

1.2

The aim and objectives in the study .................................................................. 2

CHAPTER 2. LITERATURE REVIEW ................................................................. 4
2.1

Landscape lenses ............................................................................................... 5

2.2

Achromatic landscape lens ............................................................................... 6


2.3

The Petzval Portrait lens ................................................................................... 8

2.3.1

Rapid Rectilinear lens .............................................................................. 10

2.4

The Cooke lens................................................................................................ 11

2.5

The Celor lens ................................................................................................. 13

2.6

The Tessar lens................................................................................................ 15

CHAPTER 3. USEFUL OPTIC THEORIES AND METHODS .......................... 17
3.1

Useful optic theories ....................................................................................... 17

3.1.1

The derivation of primary axial color equation ....................................... 17

3.1.2


Field curvature flattening ......................................................................... 20

3.1.3

Power equation for multiple elements optical system ............................. 21

3.1.4

Celor equation derivation and apply for Cooke triplet design ................. 23

3.2

Methods ........................................................................................................... 25

3.2.1

Cooke Triplet lens and specifications ...................................................... 25

3.2.2

How to select the glasses.......................................................................... 26

3.2.3

Design procedure of Cooke triplet ........................................................... 28

3.2.4

Tessar lens and specification .................................................................... 32


CHAPTER 4. RESULTS ....................................................................................... 35
4.1

The Cooke Triplet lens .................................................................................... 35

4.1.1

The progression ........................................................................................ 35
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4.1.2
4.2

Final results and comparison .................................................................... 42

Tessar lens f/2.8, 52mm .................................................................................. 48

4.2.1

Lens data .................................................................................................. 48

4.2.2

System descriptions .................................................................................. 49


4.2.3

Layout of lens ........................................................................................... 50

4.2.4

Ray fan plots............................................................................................. 51

4.2.5

Field curvature and distortion .................................................................. 51

4.2.6

Modulation transfer function.................................................................... 52

4.2.7

Image simulation ...................................................................................... 53

Chapter 5. Conclusions .......................................................................................... 55

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TABLE OF FIGURES
Figure 1.1 Photographic lens illustration........................................................................ 1
Figure 1.2. Flowchart diagram ....................................................................................... 3
Figure 2.1 The classification of lens............................................................................... 4
Figure 2.2 System descriptions and 3D layout ............................................................... 5
Figure 2.3 Distortions and field curvature of lens .......................................................... 6
Figure 2.4 The Landscape lens design by Zemax .......................................................... 6
Figure 2.5 Achromatic lens design by Zemax ................................................................ 7
Figure 2.6 Spot diagram, achromatic focal shift ............................................................ 8
Figure 2.7 Petzval lens design by Zemax ....................................................................... 9
Figure 2.8 Petzval lens design by Zemax ....................................................................... 9
Figure 2.9 Spot diagram, ray fan plot of Petzval portrait lens ..................................... 10
Figure 2.10 Rapid Rectilinear lens ............................................................................... 10
Figure 2.11 Field curvature and distortion, achromatic focal shift .............................. 11
Figure 2.12 Doublet (a), separated of doublet (b) ........................................................ 12
Figure 2.13 Cooke Triplet (left), H. Dennis Taylor (right) .......................................... 13
Figure 2.14 The Celor lens ........................................................................................... 14
Figure 2.15 Spot diagram of the Celor lens .................................................................. 15
Figure 2.16 The layout of Tessar lens .......................................................................... 15
Figure 3.1 Chromatic aberration ................................................................................... 17
Figure 3.2 Primary axial color ...................................................................................... 18
Figure 3.3 Marginal rays and thin lens ......................................................................... 19
Figure 3.4 Two elements optical system ...................................................................... 21
Figure 3.5 Interactive Abbe-Diagram........................................................................... 27
Figure 3.6 Illustration image of Cooke Triplet ............................................................. 28
Figure 3.7 Rear half of Cooke Triplet .......................................................................... 29
Figure 3.8 Cooke Triplet design procedure .................................................................. 31
Figure 3.9 illustration of Cooke Triplet lens ................................................................ 33
Figure 3.10 illustration of Tessar lens .......................................................................... 34
Figure 4.1 Rear half of Cooke Triplet design using Zemax ......................................... 35

Figure 4.2 Cooke triplet 52mm, f/5 design using Zemax ............................................. 37
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Figure 4.3. Modulation transfer function after hammer optimization (MTF1) ............ 38
Figure 4.4 Modulation transfer function (MTF2)......................................................... 40
Figure 4.5 . Modulation transfer function (MTF3) ...................................................... 40
Figure 4.6 Modulation transfer function (MTF4)......................................................... 41
Figure 4.7 Spot diagram, modulation transfer function (MTF5) ................................. 41
Figure 4.8 System data ................................................................................................. 42
Figure 4.9 3D-layout of Cooke Triplet lens ................................................................. 43
Figure 4.10 Transverse ray fan plot .............................................................................. 44
Figure 4.11 Field curvature and distortion ................................................................... 44
Figure 4.12 Modulation transfer function (MTF) ........................................................ 45
Figure 4.13 Image simulation ....................................................................................... 46
Figure 4.14 System descriptions ................................................................................... 49
Figure 4.15 3D-layout of lens ....................................................................................... 50
Figure 4.16 Transverse ray fan plot .............................................................................. 51
Figure 4.17 field curvature and distortion .................................................................... 52
Figure 4.18 Modulation transfer function .................................................................... 52
Figure 4.19 Image simulation ....................................................................................... 53
Figure 5.1 Double Gauss lens ....................................................................................... 56
Figure 5.2 Telephoto lens ............................................................................................. 56
Figure 5.3 Fisheye lens ................................................................................................. 56

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LIST OF TABLES
Table 3-1 target of Cooke Triplet design to approach .................................................. 26
Table 3-2 Glasses information ...................................................................................... 28
Table 3-3 specification of Tessar lens .......................................................................... 32
Table 4-1 Variables use in Merit function .................................................................... 36
Table 4-2 Operands use in the design........................................................................... 39
Table 4-3 Triplet Lens data .......................................................................................... 46
Table 4-4 Aspheric data ................................................................................................ 46
Table 4-5 Cooke Triplet specifications requirement and result ................................... 47
Table 4-6 Tessar lens data ............................................................................................ 48
Table 4-7 Aspheric surface data ................................................................................... 49
Table 4-8 Tessar lens specifications requirement and result ........................................ 54

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Chapter 1: INTRODUCTION
1.1

Review of photographic lens


Photographic lens (also known as camera lens) is an optical system which was the
combination of several lens elements used in conjunction with a camera body and
mechanism to make images of objects either on photographic film or on other media
capable of storing an image chemically or electronically (figure 1.1). Each of lens
element in the system directs the path of light rays from the object to recreate the image
as accuracy as possible.

Figure 1.1 Photographic lens illustration
( />
Any of photographic lens was suffered from some kind of optical aberrations that
directly affect to the quality of the final image. Optical aberrations occur when points
in the object do not translate back onto single points after passing through the lens causing image blurring, reduced contrast or misalignment of colors. That is why in the
development history of camera lens, the inventors was spent a lot of their time to
researched, then created a new type of lens that could eliminated the aberrations. Until
now, the fighting with aberrations in design photographic lens is still on the go. The two
most important properties in any photographic lens system is the effective focal length
and aperture speed (f-number). While the focal length decided the angle field of view,
the f-number will affect to the time to take a picture. For demands, people more and
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more interested in a fast camera lens, this mean the f-number must become large. This
made the task of design a new lens become more difficult because the larger f-number
the lens system undergoes the more of aberration.
Nowadays, with the development of technology and the computer is more powerful,

design photographic lens using software become much easier and save a lot of time.
There are many softwares were created for optical design purposes such as Code V,
Zemax, Winlens, OSLO, etc. In this study, Zemax was used. It is an outstanding design
and simulation software that allows scientists, engineers, researchers and students to
turn their optical and illumination systems ideas into reality.
1.2

The aim and objectives in the study

As mentioned above, the objective here was the photographic lens (camera lens). To
get a more understanding about the history of photographic lens, chapter 2 will be the
review of several important classic lenses. Each type of lens will be redesigned using
Zemax. Then the lens’s properties will be carefully analyzed to point out the pros and
cons such as what kind of aberrations it was suffered from and what is the weaknesses
in the present lens. Then we can understand the demands of creating new type of lens
that was the solution of the previous one. This is the first aim in this study.
The Cooke Triplet lens and the Tessar lens were the two famous photographic lenses
which were considered as the revolutions. These lenses as well as their modifications
are now widely used in many cameras and being the commercial products on the market.
So, designing the new versions Cooke Triplet and the Tessar lens will be the second
aim. Every aspects of the design were showed in the chapter 4.
Along with the aims of the study, some important optical theories will be mentioned
also. They are very useful theories that will be applied to the photographic lens design.
The method that was used to design two new versions Cooke Triplet and Tessar lens
was exhibited in chapter 3. The whole content has been discussed through this thesis
could see clearly from the flowchart diagram below.

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photographic
lens

history

achromatic

Petzval
sum

review several
classic lenses

power and
Celor
equation

optic theories
and methods

predescription

Cooke Triplet
lens

analyze


Tessar lens

calculation
and design

redesign
with zemax

analyzes
properties

predescription

analyze

calculation
and design
conclusions

Figure 1.2. Flowchart diagram

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CHAPTER 2. LITERATURE REVIEW

In present chapter, we are going to review several important classic lenses. Our works
will follow the order of the figure 2.1 below. It is very clear and easy to understand. The
figure lists out the name, the number of element in each lens, which element is positive
and which one is negative. The time when it was invented also presented. Time flow
will be our direction to work out but not the number of element. Each of lens will be
redesigned, analyzed the advantages as well as the drawbacks. The Landscape lens,
Achromat Landscape lens, Petzval portrait lens, Rapid rectilinear lens and Celor lens
are the main topics to be discussed in this chapter.

Figure 2.1 The classification of lens

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2.1 Landscape lenses
The earliest photographs were made by placing paper covered with a light-sensitive
material in the focal plane of a camera obscura, the lenses used being the first simple
plano-convex lenses, and later simple meniscus landscape lenses as suggested by
Wollaston in 1812 (Fig 2.2). A suitably designed meniscus lens, with a stop in front of
it on the concave side of the lens, will give good pictures at f/11 or f/16, covering with
moderate definition a total field of about 450. This lens is still universally adopted in
low-priced cameras. In addition to its cheapness, this lens has the advantage of
possessing only two glass air surfaces. (By Mc Graw-Hill Book Company, Inc).

Figure 2.2 System descriptions and 3D layout


The Landscape lens design by Zemax

There are two different versions of the land-scape lens: with the stop in front or with
the stop behind the lens, considered from the object location. Both setups are nearly
equivalent in correction, but the stop in front version is more desirable, because of the
resulting more familiar barrel distortion. The bending effect of the lens is optimized to
obtain a proper correction of astigmatism. It is usual to flatten the tangential image
surface, the astigmatism can-not be fully corrected, and the sagittal image surface is
rather poorly corrected. The bending is not optimized to correct the spherical aberration,
therefore only small apertures can be used.
Figure 2.2 shows a landscape lens with a stop in front of the lens in 3D layout. The focal
length is f = 400 mm, F-number is 15, full field of view is 2 x 250 in spectral range of
F, d, C lights. It also shows the distortion (Fig 2.3), which is of the barrel type and has
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a maximum value of 2.7%. The tangential image surface is better than sagittal one.
Figure 2.4 shows the corresponding spot diagrams for three fields. A serious achromatic
aberration was presented, the lack of this lens.

Figure 2.3 Distortions and field curvature of lens

The Landscape lens design by Zemax

Figure 2.4 The Landscape lens design by Zemax


Spot diagram, chromatic focal shift windows

2.2 Achromatic landscape lens
The lack of achromatism of the simple meniscus landscape lens was soon found to be a
disadvantage, even before the camera obscura became a photographic camera, and the
achromatic landscape lens was introduced by Chevalier in 1821 (Fig 2.5). The process
of achromatization automatically removed both of the chromatic aberrations, thus
improving the definition in a twofold manner.
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The lens was constructed as a cemented doublet, color correction and a better correction
of the spherical aberration can be performed. If the negative flint lens is located towards
the stop and the stop has a distance of nearly 1/5 of the focal length and is oriented to
the object side, the lens is named after its invertor: a Chevalier or French landscape lens.
Another well-known setup due to Grubb has a similar stop arrangement, but a positive
crown lens is located close to the stop. Figure 2.5 shows a simple example of Grubbtype achromatic lens in 3D which has an effective focal length 400mm, f-number is 16
with the stop in front of the positive lens. The ray fan plot, distortion and field curvature
also are presented.

Figure 2.5 Achromatic lens design by Zemax

System data, ray fan plot, 3D view, distortion and field curvature of the
achromatic lens

From the ray fan plot we can see that the spherical aberration of on axis field is better

than landscape lens. Let’s take a look at the spot diagram (Fig 2.6), it clearly has a
smaller diameter that expresses a better performance. Besides that, the chromatic focal
shift presents the correction of chromatic aberration.

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Figure 2.6 Spot diagram, achromatic focal shift

2.3 The Petzval Portrait lens
The Landscape lens at f/11 was successfully adopted in the early daguerreotype process,
but exposures of half an hour or more were necessary even in sunlight. Consequently
when daguerreotype portraiture was attempted the need for a much faster lens. J. Petzval,
of Vienna, solved the problem in 1841 by the design of his well-known portrait lens.
While doing portraits, people tended to fidget if exposure times exceeded a few seconds.
The result is some blurring of the image due to object movement. This does not help
customer satisfaction (Introduction to lens design: with practical Zemax example, by
Geary, Joseph M). The new industry wanted and needed a faster lens, hence they held
a contest. Petzval’s design was an f/3.6 (figure 2.7), about twenty times faster than the
lenses then on the market.
The setup for a Petzval portrait lens consists of two part with the stop lying between
them. In the optimization process, we have to satisfy two important design constraints.
The first is that the stop will be held midway between the two lens groups. This is done
by slaving the rear-stop airspace to the front-stop airspace. The second constraint is that
the ratio of the system focal length to the back focal length be equal to two. In the
example, we are going to design a 125mm EFL, f/5, cover a field of 200. In comparison

with the landscape lens, it is clearly that the Petzval lens shows rather better
performance. With the F-number of 5 it is a much faster lens. From the figure 2.7 we
can see the 3d view as well as the miscellaneous system data of the lens.

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Figure 2.7 Petzval lens design by Zemax

System data, 3D-layout of lens

Besides that, the chromatic focal shift plot (figure 2.8) presented the correction of
chromatic aberration, both of color aberrations has been corrected significantly. The
field curvature is just approximately 0.05 inch and distortion of 0.9%.

Figure 2.8 Petzval lens design by Zemax

Field curvature and distortion, chromatic focal shift graph

From the ray fan plot we can see that the spherical aberration of on axis field is better
than that of landscape lens. Let’s take a look at the spot diagram (Fig 2.9), diameters
are smaller compare to that of landscape lens that express a better performance. Notably,
the Petzval portrait lens is not a symmetrical. It suffered from the disadvantage of
astigmatic defects in the outer part of the field, which could not be removed so long as
the designer was limited to the use of ordinary crown and flint glasses.


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Figure 2.9 Spot diagram, ray fan plot of Petzval portrait lens

2.3.1 Rapid Rectilinear lens

In 1866, Dllmeyer and Steinheil simultaneously and independently realized that if two
identical lenses are mounted symmetrically about the central stop, the three transverse
aberration-distortion, chromatic difference of magnification, and coma, are
automatically removed and hence each component of such a symmetrical system need
not be corrected for any of these aberrations. They therefore constructed a symmetrical
lens, each half of which was corrected for longitudinal chromatic and spherical
aberration; the astigmatism was then removed by placing the stop at the correct position
relative to each component. In this way they produced the well-known Rapid Rectilinear
or Aplanat lens (Fig 2.10).

Figure 2.10 Rapid Rectilinear lens

System description, 3D-layout of the symmetrical lens

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It is a symmetric achromat which made use of two glasses whose Abbe numbers were
not that far apart. This allows for a more compact wide field lens design, and a flatter
illumination field. In our design example, LF1 and F1 glasses were used. The lens has
an effective focal length of 254mm, f/8 and cover a field of 400. Also, the figure 2.11
points out that the problem of distortion was improved, just 0.05%. The color correction
was guaranteed.
From this point of design process, if we continue optimize the lens system by breaking
the symmetry of the lens, it can be better. This is a reasonable thing to do when the
object-image conjugates are themselves unsymmetrical.

Figure 2.11 Field curvature and distortion, achromatic focal shift

2.4 The Cooke lens
Since it was born in early of 19th century, the camera development progress has
undergone several revolutions to obtained achievement today. Among these was the
famous invention by H. Dennis Taylor in 1893 of the Cooke Triplet lens.
As optical manager of T. Cooke & Sons of York, makers of astronomical telescopes,
H. Dennis Taylor attempted to eliminate the optical distortion or aberration at the outer
edge of lenses. In 1893 he designed and patented the revolutionary, and now famous,
Triplet design (British patent no. 1991). His work was done in algebra and calculus,
without trigonometry. Taylor’s methods were to do preliminary calculations, fabricate
and a prototype lens, test it to find its good and bad points, then go back to his
calculations and tweak his design.

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The major problem with the lenses of the time remained astigmatism and field curvature
near the edges of the image. Taylor realized that if he took a thin positive and thin
negative lens of the same power and placed them in contact (Figure 2.5a), they would
have a zero Petzval sum (meaning they would have a perfectly flat field). It would also
have zero power, meaning it didn’t magnify or focus the image. If he separated the two
elements (Figure 2.5b), however, the lens would begin to have significant positive
power, but would retain a zero Petzval sum, meaning the field of focus would still not
be curved.

Figure 2.12 Doublet (a), separated of doublet (b)

There was a problem, however. Most lenses of the day were symmetrical about a central
stop (aperture), because such symmetry eliminated a lot of aberrations. The
unsymmetrical lens would have horrible lateral aberrations. Taylor determined that if
he split one of the elements in two, mounting each of the split elements on either side
of the remaining element, the lens would be more symmetric and the aberrations
decreased. He patented both possible combinations in 1893, but preferred the negative
element in the center, surrounded by the two halves of the positive element.
The Triplet lens consists of a negative lens placed between two positive lenses. The
negative lens using flint glass which has low Abbe number or high dispersion. Vice
versa, the positive uses the crow glass with low dispersion. Using smallest number of
elements, having 14 degree of freedom (six curvatures, three lens thickness, three glass
types, two airspaces), it was the first photographic lens that allowed the reduction of the
third order aberrations (spherical, coma, astigmatism, field curvature and distortion) and
the first order one (axial color chromatic, lateral chromatic) to a value close to zero.

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Figure 2.13 Cooke Triplet (left), H. Dennis Taylor (right)

This was a successful design, the original aperture was f/4.5, but this was increased to
f/2.8. Then by splitting and/or compounding the three elements a wide variety of triplet
derivatives were produced. The Zeiss Tessar lens, designed independently in 1902 by
Rudolph and in production ever since, is one of the best known of these.
Being different, the Triplet allowed a different set of modifications. Over 80 patents
have been issued for variations and modifications of the Cooke Triplet, more than for
any other type of lens. Nowadays, with the aid of computer software, achievements of
science and technology, especially the advancement in coating technique, it allows us
to design and develop a more and more excellent lens with higher quality base on Cooke
Triplet design.
2.5 The Celor lens
In connection with the Cooke lens above, it was mentioned that the Petzval sum can be
reduced by separating the positive and negative element of an achromatic doublet. If
two such separated doublets are mounted symmetrically about a central stop, a lens is
obtained which offers even more possibilities for a good design that does the Cooke
lens. Two independent series of designs based on this general principle have been
developed, one in which the four lenses are all biconvex or biconcave and the other in
which all four lenses are meniscus-shaped. The first form is exemplified by the Goerz
Celor f/4.5, designed by von Hoegh in 1898 (Fig 2.22). Later modifications of this type
are the Goerz Dogmar, the Steinheil Unofocal, and the Taylor-Hobson Aviar (Mc GrawHill Book Company, Inc).
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Figure 2.14 The Celor lens

System descriptions, field curvature and distortion, chromatic focal shift
and the 3D layout

Here, we will do an example design of Celor lens using Zemax. The lens has 125 mm
effective focal length with f-number of 6, using BAF4 and SK4 glasses for F, d, C
spectrum, cover a field of view of 400. It is a symmetric lens with a pair of air-spaced
achromats. The air-space provides a new degree of freedom, means that the individual
lens powers must increase to preserve the color correction. This can also be done in
such a way as to lower the Petzval sum. Besides that, try to control spherical aberration
and astigmatism in the design process. The symmetry of the lens system helps to reduce
the coma, distortion and lateral color. Figure 2.23 show the spot diagram which has a
small dimension compare to previous type of lens.

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Figure 2.15 Spot diagram of the Celor lens

2.6 The Tessar lens

By cementing together the rear elements of an unsymmetrical Celor-type lens, Rudolph
in 1902 produced the Tessar lens (Fig 2.24) which is probably the best known and most
generally used type of lens produced in recent times. Tessar comprises four elements in
three groups, one positive crown glass element at the front, one negative flint glass
element at the center and a negative plano-concave flint glass element cemented with a
positive convex crown glass element at the rear. The airspaces are adjusted to fulfill the
Petzval sum.

Figure 2.16 The layout of Tessar lens

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