Apochromatic lens combinations: a novel design approach

Op/icr

& ,!mer

Tech/m/o,~~~. Vol. 29. No

I

1997 Elsevier

Scmce

4. pp. 217-219,

1997

Ltd. All rights reserved Printed

003tl

m Great

Bntain

3992 ‘97 $17.00 + 0.00

0030-3992(96)00060-6

PII:

ELSEVIER

Apochromatic a novel design

lens combinations: approach

P. HARI HARAN Apochromatic objectives are constructed by combining three lenses made of different materials with properly chosen powers and dispersions. This paper presents a novel approach to the design of such apochromatic combinations which automatically minimizes the residual chromatic aberration over a specific wavelength range. @ 1997 Elsevier Science Ltd.

KEYWORDS: lens combinations,

colour correction,

apochromatic

objectives

Introduction Achromatic objectives are constructed by combining two lenses made of different materials with properly chosen powers and dispersions. The usual design condition brings the focal planes for two selected wavelengths, typically the F and C lines of the solar spectrum, into coincidence.

where the values of P are taken at an appropriate intermediate wavelength.

For complete chromatic correction, the dispersions of the two materials (say, A and B) must then satisfy the condition PA -=-

VA

PB VB

It is then possible to combine two glasses in such proportions that their combination has an effective value of P which matches that of the third glass, while their effective value of V is substantially different from that for the third glass. Glasses meeting this condition can be chosen by a graphical method5.

(1)

where, for each of the materials While this design procedure has been used for many years, it has the drawback that the three wavelengths for which the focal planes are brought into coincidence are chosen arbitrarily. As a result, the procedure may not yield the optimum design for a specific wavelength range. This paper presents an alternative approach to the design of such combinations, based on a polynomial fit to the refractive index data over the wavelength range of interest. This approach automatically minimizes the residual chromatic aberration over this wavelength range.

and (3) for any arbitrarily chosen wavelength /1. This condition cannot be satisfied with conventional glasses. Better chromatic correction can be obtained with a combination of three lenses (an apochromatic combination) made of properly chosen materials, which brings the focal planes for three selected wavelengths into coincidence’m5.

Theory The equation for the focal lengthf of a thin lens made of a material of refractive index N can be written as

1l.f = mc

For a combination with unit power, the powers (+A, $B, $c) of the three lenses must satisfy the equations 44 + 4s + & = 1

where m = N - 1, c = (l/r]) - (I/@ the radii of the two surfaces.

(7) and YIand t-2 are

The availability of least-squares curve-fitting programs now makes it possible to remove the restrictions imposed by tables of partial dispersions corresponding to measurements at a fixed set of wavelengths, so that we can represent the value of m in (7) for any material A

(4)

The author is with the Department of Physical Optics, School of Physics, University of Sydney, NSW 2006, Australia. Received IO June 1996. Accepted 21 October 1996.

217

Apochromatic lens combinations:P.Hariharan

218

over a specified range of wavelengths, by a power series of the form rnA(C) =

BAg +

aA +

yAo2+ . . .

(8)

where C-J= l/A. For a combination of two thin lenses described by the parameters CA,mu, and cg, mB(G), respectively, the condition for achromatism can then be written in the form

cAmA + cBmB(d = 4

(9)

If we neglect terms of higher order than o2 in (S), (9) can be expanded as cB(@B+pB~+YB'J2)

Accordingly, for perfect second-order correction CANA + CB~B

=

4

(10)

chromatic

=

4

(11)

+ cBP,y =

0

(12)

CAYA -k CBYB =

0

(13)

CAPA

It follows that the materials chosen must satisfy the condition

& _

PB

(14)

YA -Kl

Table 1 gives the values of the coefficients a, JI and y in (8) and the ratio (p/Y) obtained from a polynomial fit to the available data for some typical glasses, and for fluorite6, for the range of wavelengths from 0.4046 urn to 0.7065 urn. As can be seen, the crown and flint glasses commonly used for achromatic objectives do not satisfy (14). The residual variation of the focal length with wavelength can be eliminated almost completely over a selected range of wavelengths by using three lenses made of suitably chosen materials. As outlined earlier, it is convenient to treat the system as if it consisted of only two components, but with one component consisting of two lenses made from different materials. In any such combination, the powers of two of the lenses must have the same sign, while the power of the third must have the opposite sign. Since the order of the elements in a combination of thin lenses does not affect the chromatic correction, we will assume for the present that the powers of the second and third lenses have the Table

1.

Material BSC 516642 ZC 507611 HC 524592 MBC 572577 LAF 689495 EDF 620363 Fluorite

Material

CARA + CB[MB + k&z]= CAPA

+CB[PB

CAYA + cB[YB

4

(15)

@cl= 0 i- bcl= 0

(16)

+

(17)

where k = ccfce

(18)

and the condition for chromatic correction becomes

where I$ is the power of the combination.

cA(%4+p,@+Y~g~)+

same sign. Equations (11) to (13) can then be rewritten in the form

PA -_=

BE+'%

YA

YB +h'c

(19)

Equation (19) can then be satisfied by choosing, for the two lenses with powers of the same sign, materials for which the values of the ratios (/I/y) bracket the value of (/3/y) for the lens whose power has the opposite sign. Since the values of these ratios are obtained by a leastsquares fit to the refractive index data over the wavelength range of interest, the residual chromatic aberration over this range is automatically minimized. As an example, we consider a design for a three-lens combination with a focal length of 1.OOOm, in which the positive lens is made of fluorite (/I/y = - 0.1975 l), while the two negative lenses are made of a medium barium crown glass (MBC 572577, p/y = -0.12694) and a lanthanum flint glass (LAF 689495, p/y = -0.40096). In this case, we have from (19) k = 0.22692

(20)

so that, from (15) and (16)

CA= + 5.535382

(21)

cB = - 1.923495

(22)

cc = - 0.436480

(23)

Figure 1 shows the residual deviations of the focal length of this combination over the range of wavelengths from 0.4046 urn to 0.7065 pm. The residual deviations for a conventional achromatic combination, corrected for the C and F lines, using a borosilicate crown glass (BSC 516642, Nd = 1.51680, V = 64.17) and an extra dense flint glass (EDF 620363, Nd = 1.62004, V = 36.37) are also presented for comparison. As can be seen, with the apochromatic combination, the second-order chromatic aberration has been eliminated by this procedure, leaving only higher-order residuals.

characteristics

c!

B

0.503316 0.494167 0.510674 0.557680 0.670672 0.605841 0.427323

+0.001312 +0.000835 -0.000212 -0.000686 - 0.003309 -0.013398 - 0.000499

Y +0.003873 +0.004131 +0.004712 +0.005402 +0.008254 +0.012738 +0.002528

Plr +0.33875 +0.20205 -0.04496 -0.12694 -0.40096 - 1.05181 -0.19751

Apochromatic

1.004

-*Apochromatic -m-Achromatic "'$5

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1 001 -

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a,

ooz-

219

P. Hariharan

is based on a least-squares polynomial fit to the refractive index data over the range of wavelengths of interest and automatically eliminates the second-order chromatic aberration over this wavelength range. This approach also opens up the possibility of minimizing higher-order residuals in multi-element systems.

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lens combinations:

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References

,

,

I

I

I

0.56

0.60

0.66

0.70

0.75

1

I

Wavelength (micrometres) Fig. 1 Residual variation of the focal length with the wavelength for an apochromatic combination using fluorite and two glasses. The results for a conventional achromatic combination are also shown for comparison

Conrady, A.E. Applied Optics and Optical Design. Oxford University Press, London (1929)

^L Kingslake, R. Lens Design Fundumentals. New York

Academic

Press.

(1978)

3

Herzberger, M., Jenkins, H. Color correction in optical and types of glass, J Opt Sot Am, 39 (I 949) 984-989

4

Herzberger, M. Modern Geometrical New York (1958)

Conclusions

5

Stephens, R.E. Selection of glasses J Opt Sot Am, 39 (1949) 398-401

A novel design approach for an apochromatic combination of three lenses is presented. This approach

6

Malitson, 1.H. A redetermination of some optical calcium fluoride, Appl Opr, 2 (1963) 1103-l 107

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