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Methodofzoomlensdesign

ARTICLEinAPPLIEDOPTICS·DECEMBER2008

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AntoninMiks

CzechTechnicalUniversityinPrague

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JiriNovak

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PavelNovak

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Method of zoom lens design

Antonín Mikš,* Jiří Novák, and Pavel NovákCzech Technical University in Prague, Faculty of Civil Engineering, Department of Physics,

Thakurova 7, 166 29 Prague 6, Czech Republic

*Corresponding author: miks@fsv.cvut.cz

Received 26 June 2008; revised 13 October 2008; accepted 13 October 2008;posted 15 October 2008 (Doc. ID 97888); published 7 November 2008

Optical systems with variable optical characteristics (zoom lenses) find broader applications in practicenowadays and methods for their design are constantly developed and improved. We describe a relativelysimple method of the design of zoom lenses using the third-order aberration theory. It presents one of thepossible approaches of obtaining the Seidel aberration coefficients of individual members of a zoom lens.The advantage of this method is that Seidel aberration coefficients of individual elements of a givenoptical system can be obtained simply by solving of a set of linear equations. By using these coefficients,one can determine residual aberrations of the optical system without detailed knowledge about thestructure of its individual elements. Furthermore, we can determine construction parameters of theoptical system, i.e., radii of curvature and thicknesses of individual elements of a given optical system.The proposed method makes it possible to determine which elements of the optical system can bedesigned as simple lenses and which elements must have a more complicated design, e.g., doubletsor triplets. © 2008 Optical Society of America

OCIS codes: 220.0220, 220.3620, 220.1000, 080.0080, 080.3620, 080.1010.

1. Introduction

Optical systems with variable optical characteristics—zoom lens systems—can be divided in two groups.One group is called transfocators and the other isvario-objective lenses. A transfocator is a telescopicoptical system with a variable magnification andit is often placed in front of the objective. A vario-objective lens is an optical system that images insuch conditions that either the object or the imageis in finite distance or the distance between the im-age and the object is finite. The main functions ofzoom lenses are to provide a continuous change ofthe focal length or magnification at almost constantand sufficient imaging quality, small residual aberra-tions [1–11], no change in the position of pupils, etc.,in the whole range of requested focal lengths or mag-nification. This change of optical characteristics isdone by the change of position (shift) of some ofthe elements of the optical system [12–22]. If theshift is chosen so as not to change the position of

the image plane in the whole range of focal lengthsor magnification, we call it a mechanical compensa-tion of the image plane position. In such optical sys-tems, it is necessary that at least one of the systemmembers is moving nonlinearly. The detailed analy-sis of zoom lenses with mechanical compensation canbe found in, e.g., [12,13,18,20–22]. If we choose theshift of optical system members to be linear, we callit an optical compensation and the position of the im-age plane changes in small increments dependent onthe construction of the optical system. The position ofthe image plane is then constant only for a few valuesof focal length or magnification. Zoom lenses withoptical compensation were analyzed in detail in,e.g., [12,15,19].

When the correction is performed on such typesof zoom optical systems, we proceed in the followingway.

First, according to the given optical characteris-tics, we have to determine the powers and axialseparations between the elements of the thin lenssystem and the position of the pupils. Then we shalldo the primary aberration analysis of individualelements of this optical system on the basis of the

0003-6935/08/326088-11$15.00/0© 2008 Optical Society of America

6088 APPLIED OPTICS / Vol. 47, No. 32 / 10 November 2008

magnitude of parameters hφ and �hφ on the particu-lar optical system member, where h is the height ofincidence of the aperture ray and �h is the height ofincidence of the principal ray on the given elementof the optical system with the power φ. This analysishas to be done for each element of a given zoom lenssystem and for various values (through the wholezoom range) of focal length or magnification of thesystem (i.e., for various positions of individualsystem members). Values hφ and �hφ determineapertures of the optical system and values h and �hdetermine the dimension of individual system mem-bers. According to the magnitude of quantities hφand �hφ, we can decide in the primary stage of thedesign process about the construction of individualoptical system members.Another step is the analysis of the optical system

using the third-order aberration theory. We often usethe third-order theory (Seidel theory) of aberrations[1–11] for design of new types of optical systems.Even though the Seidel theory is only an approxi-mate theory, its great advantage is that it providesus with a simple analytical expression of aberrationsof the optical system. Moreover, it enables us to havea deeper insight into the optical system, which is notpossible in commercial optical design software basedon numerical methods. This is important especiallyin the primary stage of the design of optical systems,where it enables us to see which properties each in-dividual element of the optical system should have toachieve the minimum value of residual aberration ofthe whole optical system. Therefore, it is possible todetermine the type of individual elements (single orcompound element) of the optical system and it re-sults in a significant simplification and accelerationof the primary stage of the design process of zoomlenses.The Seidel theory provides us with analytical

formulas for basic types of aberrations, such as sphe-rical aberration, coma, field curvature, and distor-tion. There are several possibilities for calculationof aberration coefficients that depend on the para-meters of the optical system. The design parametersare usually the curvature of surfaces, surface axialseparations, and the refractive indices of the glassesof individual members of the zoom system. We oftenstart for the first part of optical design from the thinlens model of the optical system, where we neglectthe thickness of individual lenses. The formulasfor Seidel aberration coefficients are then simplerand we are able to obtain an analytical solution insome cases (doublet, triplet, etc.). Many different setsof variables [1,2,4,6,8–11] for thin lens theory havebeen used. The formulas for Seidel aberration coeffi-cients are very complicated if we need to calculatethe refractive indices of individual lenses, even insuch a simple case as a cemented doublet [2]. In thiscase, the solution leads to the fifth-order equationand it is practically impossible to obtain the solutionfor a complex optical system.

The third step consists of the recalculation of thinlens elements to thick lens elements design, and itis followed by an optimization of the constructionparameters using some commercial optical designsoftware.

In this paper we use the modified formulas forSeidel aberration coefficients that are given in [8].By using these formulas, it is possible to calculatethe shape and the refractive index of the glass ofthe individual lenses of the optical system. Anotheradvantage is the possibility to determine whichmember of the optical system cannot be a simple lensand must be replaced by a doublet or a triplet. In ourwork, the relations for application of Seidel aberra-tion coefficients [8] for calculation of parameters ofzoom lenses (i.e., optical systems with variable opti-cal characteristics such as variable focal length andtransverse magnification) were derived. Basic infor-mation about zoom lenses can be found in [12–23].

The aim of our work is to present one possibleapproach to determine the Seidel aberration coeffi-cients of individual elements of zoom lenses. Byusing these coefficients, we can determine the valuesof residual aberrations of the investigated opticalsystem without detailed knowledge of the type ofits individual elements. It also enables us to deter-mine the construction parameters of the optical sys-tem, i.e., radii of curvature and thicknesses ofindividual elements of the optical system using rela-tions described in [1,2,23–25], which is not a subjectof our paper. An optical system designed using thedescribed method then serves as a starting pointfor further optimization using software for opticalsystem design and analysis, such as ZEMAX andOSLO [26,27].

2. Imagery by a Thin Lens

For simplicity, we assume that the lens is placed inair. The conjugate equation for the thin lens in air is

1s0−1s¼ 1

f 0¼ φ; ð1Þ

where s is the object distance, s0 is the imagedistance, f 0 is the focal length of the lens, and φ isthe power of the lens. For the power φ, we have

φ ¼ ðn − 1Þ�1r−1r0

�; ð2Þ

where n is the refractive index of the glass fromwhich the lens is made and r and r0 are the radiiof curvature of the lens. The magnification m canbe expressed as

m ¼ s0

s¼ 1

1þ sφ : ð3Þ

We now define two parameters [2] (variables), X andY . The shape of the lens can be characterized byso-called shape parameter X. The shape parameter

10 November 2008 / Vol. 47, No. 32 / APPLIED OPTICS 6089

X (bending factor) of the thin lens is defined as

X ¼ r0 þ rr0 − r

: ð4Þ

Parameter Y is called the conjugate parameter and isdefined as

Y ¼ s0 þ ss0 − s

: ð5Þ

We can see that, for both radii r and r0, there is onlyone shape parameter X. Knowing the lens power φ,its refractive index n, and the shape parameter X,we can calculate the radii of curvature of the lens.Using Eqs. (2) and (4), we have

r ¼ 2ðn − 1ÞφðX þ 1Þ ; r0 ¼ 2ðn − 1Þ

φðX − 1Þ : ð6Þ

For the conjugate parameter Y, we have

Y ¼ s0 þ ss0 − s

¼ mþ 1m − 1

¼ −1 −2sφ ¼ 1 −

2s0φ : ð7Þ

3. Seidel Aberration Coefficients

Assume a thin lens optical system in air that consistsof K lenses. Seidel aberration coefficients [1,2,8](Seidel sums) SI, SII, SIII, SIV , and SV are givenby the following formulas:

SI ¼XKi¼1

h4i Mi; ð8Þ

SII ¼XKi¼1

h3i�hiMi þ

XKi¼1

h2i Ni; ð9Þ

SIII ¼XKi¼1

h2i�h2i Mi þ 2

XKi¼1

hi�hiNi þ

XKi¼1

φi; ð10Þ

SIV ¼XKi¼1

φi

ni; ð11Þ

SV ¼XKi¼1

hi�h3i Mi þ 3

XKi¼1

�h2i Ni þ

XKi¼1

�hi

hi

�3þ 1

ni

�φi;

ð12Þ

where

Mi ¼ φ3i ðAiX2

i þ BiXiYi þ CiY2i þDiÞ;

Ni ¼ φ2i ðEiXi þ FiYiÞ; Ai ¼

ni þ 2

4niðni − 1Þ2 ;

Bi ¼ni þ 1

niðni − 1Þ ; Ci ¼3ni þ 24ni

; ð13Þ

Di ¼n2i

4ðni − 1Þ2 ; Ei ¼ Bi=2; Fi ¼2ni þ 12ni

;

φi ¼ ðni − 1Þ�1ri−1r0i

�¼ 1

s0i−1si; Xi ¼

r0i þ rir0i − ri

;

Yi ¼s0i þ sis0i − si

¼ mi þ 1mi − 1

¼ −1 −2

siφi¼ 1 −

2s0iφi

;

Yiþ1 ¼ hiφi

hiþ1φiþ1ðYi − 1Þ − 1: ð14Þ

In previous equations, we denoted hi as the incidenceheight of a “paraxial aperture ray” (auxiliary aper-ture ray) at the ith lens, �hi as the incidence heightof a “paraxial principal ray” (auxiliary principalray) at the ith lens, ri and r0i as the radii of the ithlens, si and s0i as the object and image distances fromthe ith lens, ni as the refractive index of the ith lens,φi as the power of the ith lens, SI as the Seidel sumfor spherical aberration, SII as the Seidel sum forcoma, SIII as the Seidel sum for astigmatism, SIVas the Seidel sum for field curvature (Petzvalsum), and SV as the Seidel sum for distortion.

We can calculate the radii of curvature of the ithlens in the case where we know its lens power φi, re-fractive index ni, and shape parameter Xi. By usingEq. (6), we obtain

ri ¼2ðni − 1ÞφiðXi þ 1Þ ; r0i ¼

2ðni − 1ÞφiðXi − 1Þ :

Transverse ray aberrations δy0 and δx0 in the imageplane of the optical system that is composed ofK thinlenses in air can be then calculated from the follow-ing equations:

δy0 ¼ −yPðy2P þ x2PÞ

2ðs1 − �s1Þ3u31u

0K

SI þy1ð3y2P þ x2PÞ

2ðs1 − �s1Þ3u21u

0K �u1

SII

−y21yP

2ðs1 − �s1Þ3u1u0K �u

21

ð3SIII þ I2SIVÞ

þ y312ðs1 − �s1Þ3u0

K �u31

SV ;

δx0 ¼ −xPðy2P þ x2PÞ

2ðs1 − �s1Þ3u31u

0K

SI þ2y1yPxP

2ðs1 − �s1Þ3u21u

0K �u1

SII

−y21xP

2ðs1 − �s1Þ3u1u0K �u

21

ðSIII þ I2SIVÞ;

6090 APPLIED OPTICS / Vol. 47, No. 32 / 10 November 2008

where xP and yP are coordinates of the ray in theplane of entrance pupil of the optical system, y1 isthe distance of the object point from the optical axis,s1 is the distance of the object plane from the firstsurface of the optical system, �s1 is the entrance pupildistance from the first lens of the optical system,u1 ¼ h1=s1 is the angle of the paraxial apertureray in the object space, �u1 ¼ �h1=�s1 is the angle ofthe principal ray in the object space, u0

K ¼ u1=m isthe angle of the paraxial aperture ray in the imagespace, and m is the transverse magnification of theoptical system. Furthermore, we denoted I as theLagrange–Helmholtz invariant, which is definedfor the optical system in air as

I ¼ h1�h1

�1s1

−1�s1

�¼ u1

�h1 − �u1h1:

Without loss of generality, we can put theLagrange_Helmholtz invariant I ¼ 1 and h1 ¼ 1;then we have

�h1 ¼ s1�s1�s1 − s1

:

The Seidel difference formula (for I ¼ 1 and h1 ¼ 1)between h and �h is given by

�hj ¼ hj

��h1 þ

Xj

i¼2

di−1

hi−1hi

�;

where di denotes the axial separation between theith and ðiþ 1Þth lenses.

4. Modification of the Formulas for Third OrderAberration Coefficients

The aberration coefficients [8] are quadratic func-tions in X and Y . Their dependence on the refractiveindex is very complicated. It is a very difficult pro-blem to obtain a solution for X and n for the complexoptical system because we must solve several non-linear equations. To obtain the solution for X andn for a complex optical system, we define new setof variables. If we put φi ¼ 1 and Yi ¼ 0 (without lossof generality) in Eq. (13), we can define newparameters:

�Mi ¼ Miðφi ¼ 1;mi ¼ −1Þ;�Ni ¼ Niðφi ¼ 1;mi ¼ −1Þ;�Mi ¼ AiX2

i þDi; �Ni ¼ EiXi: ð15Þ

The parameters �M and �N describe the sphericalaberration and the coma for the thin lens having unitfocal length and unit magnification (m ¼ −1). Weobtain, for the spherical aberration,

δs0 ¼ −2H2 �M;

and for the sine condition

δm ¼ H2ð �M þ �N=2Þ;

where H is the incidence height of the aperture ray.Other variants of aberration coefficients are given inRef. [8]. Upon substituting Eq. (14) into Eq. (15), wefind that

�Mi ¼14

�ni þ 2

niðni − 1Þ2 X2i þ

n2i

ðni − 1Þ2�; ð16Þ

�Ni ¼12

�ni þ 1

niðni − 1ÞXi

�: ð17Þ

These new parameters depend only on the shapeparameter X and the index of refraction n of the lensand do not depend on the focal length of the lens.Upon substituting Eq. (17) into Eq. (16), we find that

�Mi ¼ Di þAi

E2i

�N2i ;

where

Di ¼n2i

4ðni − 1Þ2 ;Ai

E2i

¼ niðni þ 2Þðni þ 1Þ2 :

As we can see, the value of the expression A=E2 doesnot change significantly in relation to n, as isapparent from the example shown below:

n ¼ 1:5 ⇒ A=E2 ¼ 0:84; n ¼ 2:0 ⇒ A=E2 ¼ 0:89:

We can then consider this expression as a constantand put approximately A=E2

≈ 0:86. Then we obtain

�Mi ¼ Di þ 0:86 �N2i : ð18Þ

As one can see, values of expressions forC and F varyvery slowly with the value n. We can again considerthese expressions as constants and put C ≈ 1:06 andF ≈ 1:31 (considering the average value of the indexof refraction approximately n ¼ 1:6). Upon substitut-ing Eqs. (16) and (17) into Eq. (13), we find that itholds approximately

Mi ¼ φ3i ð �Mi þ 2 �NiYi þ 1:06Y2

i Þ; ð19Þ

Ni ¼ φ2i ð �Ni þ 1:31YiÞ; ð20Þ

�Mi ¼ f 03i Mi − 2f 02i NiYi þ 1:56Y2i ; ð21Þ

�Ni ¼ f 02i Ni − 1:31Yi: ð22Þ

10 November 2008 / Vol. 47, No. 32 / APPLIED OPTICS 6091

These formulas are sufficiently accurate for all prac-tical cases. As we can see, Eqs. (19) and (20) are lin-ear in parameters �M and �N. The Seidel sums are alsolinear in these parameters. We have the possibility toobtain not only shapes of the lenses, but also the re-fractive indices of the glasses from which the lensesare made.The following steps are used in a third-order aber-

ration design of a lens system. First, we choose whichSeidel sum must be corrected and then we solve thesystem of the linear equations in �M and �N. We thenobtain the values of �M and �N for the individuallenses. Upon substitution for �M and �N intoEq. (18), we have

D ¼ �M − 0:86 �N2: ð23Þ

For the refractive index of the lens we have [Eq. (14)]

n ¼ffiffiffiffiD

pffiffiffiffiD

p− 0:5

: ð24Þ

The real value for the refractive index (1:43 < n < 2)we obtain 1 < D < 2:76. In the case when the value ofD exceeds the above-mentioned bounds, we mustreplace the simple lens by a more complex opticalsystem, for instance, a cemented doublet or triplet.The shape parameter X we determine fromEq. (16) or Eq. (17). We can calculate the radii oflenses using Eq. (6), repeated here:

r ¼ 2ðn − 1ÞφðX þ 1Þ ; r0 ¼ 2ðn − 1Þ

φðX − 1Þ :

From Eq. (7), we determine the object and imagedistances from

1s¼ φ

2ð−Y − 1Þ; 1

s0¼ φ

2ð−Y þ 1Þ: ð25Þ

When designing an optical system in such a way, wechoose the glasses for some lenses and for the re-maining lenses we determine the refractive indicesas described above. This situation frequently ap-pears in practice. In the case when the element ofthe optical system is composed of several thinelements in contact (compound element), then ourformulas are also valid. It is only necessary to under-stand parameters �M and �N as parameters that char-acterize spherical aberration and the coma of thewhole compound element (e.g., cemented doublet, tri-plet). By calculation of parameters �M and �N, we candetermine the spherical aberration and the coma ofthe compound element and, subsequently, we cansimply calculate the radii and refraction indices ofindividual lenses. There exist several works, e.g.,[1,2,23–25], where the formulas for such calculationare given. The detailed description of the calculationof parameters (radii of curvature and axial distancesbetween elements) of various types of compound

optical elements with given values of Seidel aberra-tion coefficients can be found especially in [1,2,23],where both the necessary theoretical relations andnumerical examples of calculations of such opticalsystems are provided. As we can verify by directcalculation, the approximations used give almostthe same results as exact calculation.

5. Seidel Aberration Coefficients for Zoom LensDesign

If we substitute Eqs. (19) and (20) into Eqs. (8)–(12),we obtain the following formulas for Seidelaberration coefficients:

SIj ¼XKi¼1

h4jiφ3

i�Mi þ 2

XKi¼1

h4jiφ3

i Yji�Ni

þ 1:06XKi¼1

h4jiφ3

i Y2ji;

SIIj ¼XKi¼1

h3ji�hjiφ3

i�Mi þ

XKi¼1

h2jiφ2

i ð2hji�hjiφiYji þ 1Þ �Ni

þXKi¼1

h2jiφ2

i Yjið1:06hji�hjiφiYji þ 1:31Þ;

SIIIj ¼XKi¼1

h2ji�h2jiφ3

i�Mi

þ 2XKi¼1

hji�hjiφ2

i ðhji�hjiφiYji þ 1Þ �Ni

þXKi¼1

hji�hjiφ2

i Yjið1:06hji�hjiφiYji þ 2:62Þ

þXKi¼1

φi;

SIVj ¼XKi¼1

φi

ni;

SVj ¼XKi¼1

hji�h3jiφ3

i�Mi þ

XKi¼1

�h2jiφ2

i ð2hji�hjiφiYji þ 3Þ �Ni

þXKi¼1

�h2jiφ2

i Yjið1:06hji�hjiφiYji þ 3:93Þ

þXKi¼1

�hji

hji

�3þ 1

ni

�φi;

where i ¼ 1; 2;…;K, K is the number of optical sys-tem members, j ¼ 1; 2;…;L, and L is the number of

6092 APPLIED OPTICS / Vol. 47, No. 32 / 10 November 2008

chosen parameters of the zoom lens system (e.g.,values of focal lengths or magnifications, which areconsidered for correction of the optical system).Moreover, if we denote

xi ¼ �Mi;

xiþK ¼ �Ni;

aji ¼ h4jiφ3

i ;

aj;iþK ¼ 2h4jiφ3

i Yji;

aj;2Kþ1 ¼XKi¼1

1:06h4jiφ3

i Y2ji;

ajþL;i ¼ h3ji�hjiφ3

i ;

ajþL;iþK ¼ h2jiφ2

i ð2hji�hjiφiYji þ 1Þ;

ajþL;2Kþ1 ¼XKi¼1

h2jiφ2

i Yjið1:06hji�hjiφiYji þ 1:31Þ

ajþ2L;i ¼ h2ji�h2jiφ3

i ;

ajþ2L;iþK ¼ 2hji�hjiφ2

i ðhji�hjiφiYji þ 1Þ;

ajþ2L;2Kþ1 ¼XKi¼1

hji�hjiφ2

i Yjið1:06hji�hjiφiYji þ 2:62Þ

þ φi;

ajþ3L;i ¼ hji�h3jiφ3

i ;

ajþ3L;iþK ¼ �h2jiφ2

i ð2hji�hjiφiYji þ 3Þ;

ajþ3L;2Kþ1 ¼XKi¼1

�h2jiφ2

i Yjið1:06hji�hjiφiYji þ 3:93Þ

þ 3:6�hji

hjiφi;

bj ¼ SIj − aj;2Kþ1; bjþL ¼ SIIj − ajþL;2Kþ1;

bjþ2L ¼ SIIIj − ajþ2L;2Kþ1; bjþ3L ¼ SVj − ajþ3L;2Kþ1;

we can rewrite the preceding relations in matrixform. It holds that

Gx ¼ b;

where

G ¼ ðapqÞ; b ¼ ðb1; b2;…; b4LÞT ;x ¼ ð �M1; �M2;…; �MK ; �N1; �N2;…; �NKÞT ;

Starget ¼ ðSI1;SI2;…;SIL;SII1;SII2;…;SIIL;SIII1;

× SIII2;…;SIIIL;SV1;SV2;…;SVLÞT ;

whereas, for indices p and q, holds: p ¼ 1; 2;…; 4L,q ¼ 1; 2;…; 2K . Starget is the vector of required valuesof aberration coefficients (usually Starget ¼ 0). The so-lution of this linear equation system can be obtainedin the form

x ¼ �G−1b: ð26Þ

The symbol �G−1 denotes a generalized inversion ofmatrix G (e.g., for 4L > 2K holds �G−1 ¼ðGTGÞ−1GT , i.e., the solution is obtained by theleast-squares method). By solving Eq. (26), we obtainthe values of variables �M and �N for individual mem-ber of the optical system. By using Eq. (23), we candetermine which of the optical system elements canbe a simple lens and which must be composed of sev-eral lenses (doublet, triplet, etc.). When the calcula-tion is being done, we usually choose a large numberof the states of the zoom lens system, i.e., 4L ≫ 2K ,and then we calculate the values �M and �N for indi-vidual members of the zoom lens using the least-squares method. In the case we require, e.g., onlycorrection of aberration coefficients SI, SII, andSIII, then indices p and q will have values in therange p ¼ 1; 2;…; 3L, q ¼ 1; 2;…; 2K . The vectorSres of residual values of the aberration coefficientsof the optical system can be calculated from

Sres ¼ Scalc − Starget ¼ Gx − b;

where x is the solution of the linear system ofEq. (26), Scalc is the vector of calculated values ofthe aberration coefficients, and Starget is the vectorof required values of the aberration coefficients.The main advantage of our method that uses vari-ables �M and �N for the characterization of individualoptical elements is the fact that the Seidel aberrationcoefficients of individual elements of the optical sys-tem can be obtained by solving a set of linear equa-tions. In the case that we choose other variables (e.g.,radii of curvature and their various combinations),we would have to solve a set of nonlinear equations

10 November 2008 / Vol. 47, No. 32 / APPLIED OPTICS 6093

and that is a very difficult problem. Another advan-tage of this method is the fact that the values ofvariables �M and �N are always finite, which is notnecessarily true for other variables.

6. Chromatic Aberration

The refractive index is a function of wavelength forall optical media other than vacuum, an effect some-times called dispersion; since the properties of anoptical system depend on the refractive indices ofthe media, it follows that dispersion gives rise to var-iation in these properties with wavelength, Thesevariations are called chromatic aberrations [1–10].The formulas for chromatic sums for an opticalsystem of K thin lenses are given by [1,2,4,9]:

CI ¼XKi¼1

h2i

φi

νiPiλ; ð27Þ

CII ¼XKi¼1

hi�hiφi

νiPiλ; ð28Þ

where CI is the chromatic sum for a longitudinalchromatic aberration, CII is the chromatic sum fora transverse chromatic aberration, νi is the Abbenumber of the ith lens, and Piλ is the relative partialdispersion of the ith lens. Abbe number and relativepartial dispersion are defined as

ν ¼ nd − 1nF − nC

; Pλ ¼nF − nλnF − nC

; ð29Þ

where nd is the refractive index of the glass for wave-length λd ¼ 589nm, nF is the refractive index of theglass for wavelength λF ¼ 486nm, nC is the refrac-tive index of the glass for wavelength λC ¼ 656nm,and nλ is the refractive index of the glass for arbitrarywavelength λ.The sums CI and CII are usually coupled with the

Seidel sums as constituting a group of primary aber-rations. The optical system does not suffer from chro-matic aberrations if CI ¼ 0 and CII ¼ 0. We call theoptical system achromatic if chromatic aberrationsare eliminated for two wavelengths, apochromaticif chromatic aberrations are eliminated for threewavelengths, and superachromatic if chromaticaberrations are eliminated for four and more wave-lengths.In the case of a thin doublet in contact and com-

prising two different glasses or optical materials,the conditions that the doublet be corrected at twowavelengths λF and λC are

X2i¼1

φi ¼ φ;X2i¼1

φi

νi¼ 0: ð30Þ

Suitable glasses are, for example, Schott BK7/SF5(φ1 ¼ 2:0103, φ2 ¼ −1:0103). In the case of a thin

doublet in contact and comprising two differentglasses or optical materials, the conditions that thedoublet be corrected at three wavelengths λF, λC,and λd are

X2i¼1

φi ¼ φ;X2i¼1

φi

νi¼ 0;

X2i¼1

φi

νiPid ¼ 0: ð31Þ

To achieve such color correction at three wavelengths(apochromatic), the relative partial dispersion of thetwo glasses must be equal, i.e., P1d ¼ P2d. Suitableglasses are, for example, Schott FK54/KzFSN2(φ1 ¼ 2:49, φ2 ¼ −1:49). In the case of a thin tripletin contact that is composed of three different glassesor optical materials, the conditions that the triplet becolor corrected at four wavelengths (superachro-matic) λF, λC, λd, and λg are

X3i¼1

φi ¼ φ;X3i¼1

φi

νi¼ 0;

X3i¼1

φi

νiPid ¼ 0;

X3i¼1

φi

νiPig ¼ 0:

ð32Þ

Suitable glasses are, for example, Schott PK51/LaK8/FK51 (φ1 ¼ 1:7508, φ2 ¼ −2:0464, φ3 ¼1:2956).

During the design of individual members of thezoom lens system, we mostly require that theindividual member of the optical system is free ofchromatic aberration. This can be achieved byapplication of Eqs. (30)–(32) for individual types ofchromatic corrections.

By the method described in Sections 5 and 6, weobtain parameters of the optical system that isconsequently used as a starting point for furtheroptimization using commercial optical design soft-ware, such as ZEMAX or OSLO.

7. Examples

In the following subsections, we will show examplesof the application of previous formulas for the opticalsystem design. The approximate calculation will alsobe compared to calculation with respect to exactformulas.

A. Calculation of Seidel Coefficients for Thin CementedDoublet Using Exact and Approximated Formulas

To show the magnitude of errors due to usage of theapproximate formulas in Eqs. (19) and (20), we pre-sent an example of calculation of Seidel aberrationcoefficients for a cemented doublet with approximateformulas. Parameters of the cemented doublet areshown in Table 1 (linear dimensions are given inmillimeters), where r and r0 are the radii of curvatureof the lens surfaces, d is the thickness of the lenses, nis the index of refraction, and f 0 is the focal length ofthe doublet. Table 2 then presents the results of cal-culation of aberration coefficients SI and SII for thementioned doublet.

6094 APPLIED OPTICS / Vol. 47, No. 32 / 10 November 2008

In Table 2 we denoted M, N, SI, and SII as exactvalues calculated from Eqs. (13), (8), and (9) and Ma,Na, SIa, and SIIa are approximate values calculatedusing Eqs. (13), (19), and (20). As we can see fromTable 2, the performed approximation offers verygood results and it can be used in practice. It wasnoted that the main advantage of the approximationis the linearity of formulas for Seidel aberration coef-ficients with respect to parameters �M and �N. Otherexamples of applications with detailed numericaldata can be found in [8].

B. Example of Calculation of Two-Element Zoom Lens

Now, we will show the process of calculation ofparameters of the two-element zoom lens using thethird-order aberration theory. Such optical systemsare often used in practice, e.g., as image revertingsystems in riflescopes or as a part of some more com-plicated zoom lens system. The scheme of the two-element optical system is shown in Fig. 1. The systemis composed of two elements with powers φ1 and φ2and the distance between both elements is d. Theoptical system images the object in the object planeξ, which is located at the distance s1 from the firstelement of the optical system. The image is then cre-ated in the image plane ξ0, which is located at the dis-tance s02 from the second element of the opticalsystem. The paraxial aperture ray outgoing fromthe point A1 in the plane ξ intersects the first mem-ber of the optical system at height h1 from the opticalaxis A1A0

2. The paraxial principal ray outgoing fromthe point B1 in the plane ξ intersects the first mem-ber at height �h1 from the optical axis. The center ofthe entrance pupil P1 is located in the distance �s1from the first member of the optical system and inthe distance p1 ¼ �s1 − s1 from the object plane ξ.The image of the point P1 is the point P0

2 that isthe center of the exit pupil. The image plane ξ0 is si-tuated in the distance e from the object plane ξ. Theimages of points A1 and B1 are denoted as A0

2 and B02.

We obtain, for the power φ and focal length f 0 ¼1=φ of the optical system, the following equations:

φ ¼ φ1 þ φ2 − dφ1φ2; f 0 ¼ f 01f02

f 01 þ f 02 − d;

where f 01 and f 02 are values of focal length of indivi-dual elements of the optical system.We obtain, for anarbitrary point A1 that is situated in the distance s1from the first element of the system,

s1 ¼ f 0�1m

− 1þ df 02

�; s02 ¼ f 0

�1 −m −

df 01

�;

where m ¼ y02=y1 is the transverse magnification.The distance e of the image plane ξ0 from the objectplane ξ can be calculated from e ¼ −s1 þ dþ s02.

By substitution into the preceding formula, we ob-tain, for the distance d between both elements of theoptical system, the following quadratic equation:

d2− edþ ðf 01 þ f 02Þeþ f 01f

02ðm − 1Þ2=m ¼ 0:

The solution of the previous equation can be writtenas

d ¼ 12

�e�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2 − 4ðf 01 þ f 02Þe − 4f 01f

02ðm − 1Þ2=m

q �:

This formula makes it possible to calculate the dis-tance d of the two-element optical system with re-spect to the transverse magnification m of theoptical system and the distance e between the objectplane ξ and the image plane ξ0.

The process of calculation of design parameters ofa two-element zoom optical system can be performedin the following way. We choose the distance e be-tween the object and the image plane and determinefocal lengths f 01 and f 02 of both elements of the opticalsystem. The position of the entrance pupil is then gi-ven either by the distance �s1 from the first lens or bythe distance p1 from the object plane. Moreover, weset minimum and maximum values ðmmin;mmaxÞ ofthe transverse magnification m of the zoom lens,the number L of chosen parameters of the zoom lenssystem, and the required values of the aberrationcoefficients for the design process.

Using given parameters, we can calculate dis-tances s1, s02, heights h1, h2, �h1, �h2, and elementsof matrix G and vector b. By solving Eq. (26), we ob-tain the required values of variables �M1, �M2, �N1, and

Table 1. Parameters of Cemented Doublet

f 0 ¼ 1

i ri r0i di ni glass

1 0.4123 −0:8482 0 1.57488 BaK12 −0:8482 3.5013 0 1.73212 SF1

Table 2. Seidel Coefficients SI and SII of the Doublet

h1 ¼ 1, �h1 ¼ 0, m ¼ −1, s1 ¼ −2, I ¼ 1

i �Mi�Ni Mi Mia Ni Nia

1 2.0816 0.4917 16.534 16.516 −0:8154 −0:77652 1.7733 0.6571 −10:098 −10:196 3.6184 3.6436

SI ¼ 6:4364, SIa ¼ 6:2303, SII ¼ 2:803, SIIa ¼ 2:867

Fig. 1. Two-element optical system.

10 November 2008 / Vol. 47, No. 32 / APPLIED OPTICS 6095

�N2 for individual elements of the zoom lens. Fromthese values, which specify aberration properties ofindividual elements of the zoom lens, we can deter-mine [using Eq. (23)] which of the optical systemmembers can be a simple lens and which must becomposed of several lenses (e.g., doublet, triplet).Values hφ and �hφ determine apertures and valuesh, �h determine the dimension of individual systemmembers. According to the magnitude of quantities�M1, �M2, �N1, �N2, hφ, and �hφ, we can decide, in theprimary stage of the design process, about the con-struction of individual optical system members. Ifthe optical system should be achromatic, then thepower of the lenses of individual elements can bedetermined from Eqs. (27) and (28).The process of calculation of parameters of two-

element zoom lens can be carried out using thefollowing formulas:

1. Given: f 01, f02, e, �s1 (or p1), mmin, mmax, K , L.

We require, for example, correction of three typesof aberrations: spherical aberration SI ¼ 0, comaSII ¼ 0, and astigmatism SIII ¼ 0ð3L > 2KÞ.2. Calculating: m ∈ hmmin;mmaxi,

d ¼ 12

�e�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2 − 4ðf 01 þ f 02Þe − 4f 01f

02ðm − 1Þ2=m

q �;

f 0 ¼ f 01f02

f 01 þ f 02 − d; s1 ¼ f 0

�1m

− 1þ df 02

�;

s02 ¼ f 0�1 −m −

df 01

�;

h1 ¼ 1; h2 ¼ s02s1m

; �h1 ¼ s1�s1�s1 − s1

¼ s1ðp1 þ s1Þp1

;

�h2 ¼ h2�h1 þ d;

Y1 ¼ −1 −2f 01s1

; Y2 ¼ 1 −2f 02s02

; φ1 ¼ 1=f 01;

φ2 ¼ 1=f 02:

Using the previous formulas, we can calculate in-dividual elements of matrix G and vector b and wecan solve Eq. (26) with respect to x. As an exampleof the described design process, we will show calcula-tions for two-element zoom system with the followingparameters (Table 3).Given parameters:

f 01 ¼ −50mm; f 02 ¼ 50mm; e ¼ 0; �s1 ¼ 0;

mmin ¼ 0:333; mmax ¼ 1; K ¼ 2; L ¼ 9:

Required aberration coefficients:

SI ¼ SII ¼ SIII ¼ 0:

We obtain �M1 ¼ 0:734, �M2 ¼ 0:232, �N1 ¼ −0:077,and �N2 ¼ 0:102. Residual ray aberrations ðdx;dyÞof the third order that we obtain using the above-mentioned equations for the axial point (imageheight y0 ¼ 0mm) and for the numerical apertureNAmax ¼ 0:1 are shown in Fig. 2 (linear dimensionsare introduced in millimeters). As one can see, thecorrection of residual ray aberrations of the thirdorder is very good in the whole range of requiredmagnification values.

C. Example of Calculation of Three-Element Zoom Lens

As another example, we will present the calculationof the three-element zoom lens that images the objectat infinity. We designate f 0min as the minimum valueof the focal length, f 0max as the maximum value of thefocal length, f 01 as the focal length of the first ele-ment,m3 as the transverse magnification of the thirdelement of the system, s03 as the image distance, andsc as the position of the aperture stop with respect tothe third element of the optical system. Then we candetermine the remaining parameters of the opticalsystem using the following formulas:

f 03 ¼ s03=ð1 −m3Þ; f 02 ¼ −1=ð1=f 01 þ 1=f 03Þ;

d1 ¼ f 01 þ f 02 −m3f 01 f02=f

0;

d2 ¼ f 0ð1 − d1=f 01 − s03=f0Þ=m3;

where f 02 and f 03 are focal lengths of the second andthird elements of the system, d1 is the distance be-tween the first and second elements, d2 is the dis-tance between the second and third elements, andf 0 is the focal length of the optical system. Choosing,

Fig. 2. Spot diagrams of two-element zoom lens for y0 ¼ 0mmandSI ¼ SII ¼ SIII ¼ 0.

6096 APPLIED OPTICS / Vol. 47, No. 32 / 10 November 2008

for example, the following values: f 0min ¼ 60mm,f 0max ¼ 120mm, f 01 ¼ 100mm, m3 ¼ −0:4, s03 ¼70mm, and sc ¼ −40mm (aperture stop is positionedbetween the second and the third elements), we ob-tain f 02 ¼ −33:333mm and f 03 ¼ 50mm. Distances d1and d2 are given in Table 4 for various values of thefocal length f 0 (linear dimensions are introduced inmillimeters).By using previous formulas, we can calculate indi-

vidual elements of matrix G and vector b, and we cansolve Eq. (26) for K ¼ 3 and L ¼ 9 with respect to x.Required aberration coefficients are SI ¼ SII ¼SIII ¼ 0. We obtain the following parameters�M1 ¼ 0:247, �M2 ¼ 0:0154, �M3 ¼ 0:554, �N1 ¼ 0:473,�N2 ¼ −0:0089, �N3 ¼ −0:850. Residual ray aberra-tions ðdx;dyÞ of the third order that we obtain usingthe above-mentioned equations for the axial point(image height y0 ¼ 0mm) are shown in Fig. 3.Figure 4 then presents residual ray aberrations forthe off-axis point (image height y0 ¼ 14mm). Lineardimensions in Figs. 3 and 4 are given in millimeters.The f number of the optical system is 5.6. As one cansee, the correction of aberrations of the third order isvery good and stable in the whole range of requiredfocal length values.

8. Conclusion

One possible approach to obtaining Seidel aberrationcoefficients of individual elements of a zoom lens wasshown. In our method we defined new variables �Mand �N that give simple expressions for the Seidelsums. The Seidel sums are then linear functions inthese variables. The variables �M and �N depend onlyon the shape parameter X and refractive index n ofthe lens and do not depend on the focal length of thelens. The advantage of the parameters �M and �N isthe possibility to determine the refractive indicesof the lenses, which was not possible before. It is apowerful tool for design of the optical systems. Inthe case when the element of the optical system iscomposed of several thin elements in contact (com-pound element), then our formulas are also valid.It is only necessary to understand parameters �Mand �N as parameters that characterize sphericalaberration and coma of the whole compound element

(e.g., cemented doublet, triplet). By calculation ofparameters �M and �N, we can determine sphericalaberration and coma of the compound element andsubsequently we can simply calculate the radii andrefraction indices of individual lenses. It was shownon the example of a cemented doublet that the ap-proximations used give almost the same results asthe calculation using exact formulas. The same con-clusions are also valid for more complicated opticalsystems, e.g. triplets. Another advantage is the pos-sibility to determine which element of the optical sys-tem cannot be a simple lens and must be replaced bya doublet or a triplet. The formulas for calculation ofthe parameters of zoom lenses using defined vari-ables �M and �N were derived. The main advantageof our method that uses variables �M and �N for thecharacterization of individual optical elements isthe fact that the Seidel aberration coefficients of

Table 3. Parameters of the Optical System (e ¼ 0, f 01 ¼ 50mm,f 02 ¼ 50mm)

m s1 s02 d f 0 h1 h2�h1

�h2

0.333 136.607 78.866 57.741 43.297 1 1.732 0 57.7410.666 111.239 90.824 20.416 122.455 1 1.225 0 20.4161.000 100.000 100.00 0 ∞ 1 1.000 0 0

Table 4. Parameters of Three-Element Zoom Lens

f 0 d1 d2

60 44.444 91.66675 48.889 79.166105 53.968 54.167120 55.556 41.667

Fig. 3. Spot diagrams of three-element zoom lens for y0 ¼ 0mmand SI ¼ SII ¼ SIII ¼ 0.

Fig. 4. Spot diagrams of three-element zoom lens for y0 ¼ 14mmand SI ¼ SII ¼ SIII ¼ 0.

10 November 2008 / Vol. 47, No. 32 / APPLIED OPTICS 6097

individual elements of the optical system can be ob-tained by solving a set of linear equations. In the casethat we choose other variables (e.g., radii of curva-ture and their various combinations), we would haveto solve a set of nonlinear equations and that is amore difficult problem. The described process ofoptical design was shown on the example of two-element and three-element zoom lenses. As can beseen from the presented results, the proposed meth-od enables us to achieve a very good correction of rayaberrations of the third order. Parameters of an op-tical system obtained by the described method thencan be used as a starting point for further optimiza-tion using optical design software, such as ZEMAXor OSLO.

This work has been supported by grantMSM6840770022 from the Ministry of Educationof Czech Republic.

References1. B. Havelka, Geometrical Optics I, II (Czech Academy of

Science, 1955).2. D. Argentieri, Ottica Industriale (Hoepli, 1942).3. H. H. Hopkins, Wave Theory of Aberrations (Oxford, 1950).4. G. G. Slyusarev, Aberration and Optical Design Theory (Adam

Hilger, 1984).5. A. Cox, A System of Optical Design (Focal, 1964).6. P. Mouroulis and J. MacDonald, Geometrical Optics and

Optical Design (Oxford, 1997).7. A. Mikš, Applied Optics (Czech Technical U. Press, 2000).8. A. Mikš, “Modification of the formulas for third-order aberra-

tion coefficients,” J. Opt. Soc. Am. A 19, 1867–1871 (2002).9. A. E. Conrady, Applied Optics and Optical Design, Part I

(Oxford, 1929), Part II (Dover, 1960).

10. H. Haferkorn, Bewertung Optisher Systeme (VEB DeutscherVerlag der Wissenschaften, 1986).

11. W. T. Welford, Aberrations of the Symmetrical Optical Systems(Academic, 1974).

12. A. D. Clark, Zoom Lenses (Adam Hilger, 1973).13. K. Yamaji, Progress in Optics (North-Holland, 1967), Vol. VI.14. A. Wather, “Angle eikonals for a perfect zoom system,” J. Opt.

Soc. Am. A 18, 1968–1971 (2001).15. G. Wooters and E. W. Silvertooth, “Optically compensated

zoom lens,” J. Opt. Soc. Am. 55, 347–351 (1965).16. D. F. Kienholz, “The design of a zoom lens with a large

computer,” Appl. Opt. 9, 1443–1452 (1970).17. A. V. Grinkevich, “Version of an objective with variable focal

length,” J. Opt. Technol. 73 343–345 (2006).18. K. Tanaka, “Paraxial analysis of mechanically compensated

zoom lenses. 1: Four-component type,” Appl. Opt. 21, 2174–2183 (1982).

19. G. H. Matter and E. T. Luszcz, “A family of optically compen-sated zoom lenses,” Appl. Opt. 9, 844–848 (1970).

20. K. Tanaka, “Erratum: Paraxial analysis of mechanicallycompensated zoom lenses. 1: Four-component type,” Appl.Opt. 21, 3805 (1982).

21. K. Tanaka, “Paraxial analysis of mechanically compensatedzoom lenses. 2: Generalization of Yamaji type V,” Appl. Opt.21, 4045–4053 (1982).

22. K. Tanaka, “Paraxial analysis of mechanically compensatedzoom lenses. 3: Five-component type,” Appl. Opt. 22, 541–553 (1983).

23. H. Chretien, Calcul des Combinaisons Optiques (Masson,1980).

24. H. H. Hopkins and V. V. Rao, “The systematic design of twocomponent objectives,” Opt. Acta 17, 497–514 (1970).

25. M. I. Khan, “Cemented triplets: a method for rapid design,”Opt. Acta 31, 873–883 (1984).

26. www.zemax.com27. www.sinopt.com

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