Multilens system design with an athermal chart

Yasuhisa Tamagawa, Satoshi Wakabayashi, Toru Tajime, and Tsutomu Hashimoto

We have introduced an athermal chart that plots chromatic dispersive power and thermal dispersivepower on a Cartesian coordinate, and we give the design method of a multilens system in contact thatsatisfies achromatism and athermalization. The advantages of this chart are (1) that the condition ofachromatism and athermalization is clear and (2) that the approximate power of the lenses that composethe multilens system is easily found on the chart. Design indices are given through a few designexamples with an athermal chart.

Key words: Athermalization, athermal chart, optical system design.

1. Introduction

An optical system is useful for remote sensing andsurveillance systems because of its high-resolutionimage. The major factors that cause the degrada-tion of image quality are thermal aberration andchromatic aberration of optical systems. These aber-rations cause the focal-point shift of the optics,because the refractive indices of lens materials arehighly dependent on temperature and wavelength."2

To reduce these aberrations, a multilens system isused and several design methods for multilens sys-tems have been developed. Roberts dealt with ther-mal aberration and chromatic aberration separately.3He designed the achromatic optics first and thencompensated for the thermal aberration by adding achromatic aberration-free lens, such as a Ge lens inthe 8-12-jim range, but such a specific material is notalways available.

Rogers showed that the multilens system can beachromatized and athermalized with three materialsby solving three equations for lens power, achroma-tism, and athermalization,4 5 but to find a propercombination of materials, trial and error must berepeated.

To avoid time-consuming computation, graphicmethods have been developed. Gibbons presented adesign method for an athermalized doublet, using achart where the chromatic and thermal Abb6 num-bers were plotted.6 For a triplet, Rayces and Lebich

The authors are with the Electro-Optics and Microwave SystemsLaboratory, Mitsubishi Electric Corporation, 5-1-1 Ohfuna, Ka-makura, Kanagawa, Japan.

Received 28 March 1994.0003-6935/94/348009-05$06.00/0.c 1994 Optical Society of America.

introduced a yV-V diagram and showed that chro-matic aberration corresponds to the area with atriangle in the diagram.7

With these graphic methods, the power of each lensis not shown clearly on the graph. To reduce otheraberrations of multilens systems, such as the Seidelaberrations, each lens power must be as small aspossible.

In this paper, we introduce an athermal chartwhere thermal dispersive power and chromatic disper-sive power in a Cartesian coordinate are plotted.In this chart, the conditions for athermalization andachromatism are easily found, and each lens power ofthe multilens system is given graphically. In Section4, we give design indices through several numericalexamples of the multilens system.

2. Athermal Chart

Let us consider a thin lens with power i and materialMi. Then the chromatic dispersive power wi andthermal dispersive power Oi are given as follows:

oi = - i/(i= - (ani/a)AX/(ni - 1),Oi = (a-i/aT)/,i = (ni/aT)/(ni - 1) - ai,

(1)

(2)

where AX is the specified wavelength band, n is therefractive index at the center wavelength, ati is thelinear expansion coefficient of lens material Mi, and Tis the temperature. In Eqs. (1) and (2), xi and Oi areexpressed as material parameters and the wavelengthband. Therefore we can indicate the lens of materialMi by the point (i, Oi) in a Cartesian coordinate withthe axes o and 0. Here we call this chart an ather-mal chart.

First we deal with a doublet as a multilens thatconsists of lens Ll(wl, 0) and lens L2(W2, 02) in contact

1 December 1994 / Vol. 33, No. 34 / APPLIED OPTICS 8009

as shown in Fig. 1(a). Then the following equationsare obtained from the assumption of a thin lens:

= 41 + 2, (3)

° = - \/4 = (l',l + WA22)/4,

0 = (a/aT)/4 = (01,l + 022)/4l,

(4)

(5)

where 4, , and 0 are the power, chromatic dispersivepower, and thermal dispersive power of a doublet,respectively. Putting =,/= a and b2/4 = b, werewrite Eqs. (3)-(5) as

1 = a + b,

o = aol + bo)2,

(6)

(7)

.0

us

I00.cX

.-

0

.2

Chromatic dispersive power co

Fig. 2. Athermal chart for a doublet with two thin lenses.

0 = aOl + b02 - (8)

Here we putZ = X +j0,Z, = w)l +j0l, andZ2 = W2 +102 (j is an imaginary unit); the above equationsdeduce to

Z = aZl + (1- a)Z2. (9)

Equation (9) is the straight line that passes throughpoints Z, and Z2 as shown in Fig. 2. The point on theline indicates the optics with power A, which arecomposed of lenses Ll(wl, 01) and L2(W2, 02). Now,assume that wl < o2; then on the line

0 < a < 1, 0 < b < 1, when wl < < 2,

a < ,b> 1, when w > 2,

a > 1, b < 0, when w < w1 .

From Eq. (9) and Z = 0, the condition of athermal-ization and achromatism is given as follows:

01( 2 - )102 = 0-

3. Application to a Triplet

Let us consider a triplet with three thin lensesLl(0), 01), L2(w2, 02), and L3(0)3, 03). The power andmaterial of Ll, L2, and L3 are (4,, Ml), ( 2, M2), and(4N, M3), respectively. One example is shown in Fig.1(b). Now,weputZj = Wi, +jOi(i = 1,2,3).

If Zl, Z2, and Z3 are on the same line X0, thecondition of athermalization and achromatism re-duces to that of a doublet, that is, X0 must passthrough the origin.

But if Zl, Z2, and Z3 are not on the same line asshown in Fig. 3, athermalization and achromatismare always possible, because line X1, which passesthrough an appropriate point of the three points (forexample, Zl) and the origin, always intersects at pointP with the line Y that passes through the remainingtwo points (for example, Z2 and Z3), and we can real-ize the doublet corresponding to point P.

With &/4 = a, k2/4 = b, and 4h/4+ = c, thefollowing equations are obtained:

a+ b+ c = 1,(10)

dZ1 + bZ2 + jZ 3 =

(11)

(12)

.

a

.ci,

0XMn

L L LI L2 L3(a) (b)

Fig. 1. (a) Doublet with two thin lenses madematerials. (b) Triplet with three thin lensesdifferent materials.

of two differentmade of three

xi

y

Chromatic dispersive power co

Fig. 3. Method of athermalization and achromatism with threethin lenses on the athermal chart.

8010 APPLIED OPTICS / Vol. 33, No. 34 / 1 December 1994

l

3

.0OD

.U*0

0.7FEw

- h2a=hi

0

0.0

.'a

ECDi'

0Chromatic dispersive power v

Fig. 4. Graphic illustration of of a triplet on the athermal chart.

From Eqs. (11) and (12), , , and c are obtained asfollows:

_ = (0102 - 0)201) +(0203 - 302)

a ((102 - 201) + (0203 - 0)302) + (301 - 103)(13)

b = (0)102 - 0)201) (0301 - 0103)

(°102 - 201) + (0)203 - 0302) + (0)301 - 0103)

(14)

_ = (0)102 - 0)201) + ((0102 - 0201)

((0102 - 201) + (0203 - 0)302) + (0301 - 103)

(15)

The denominator indicates twice the area of thetriangle with summits Z1, Z2, and Z3, and the numera-tor of a- indicates twice the area of the trianglewith summits 0, Z2, and Z3. Therefore, as shownin Fig. 4, a is the ratio of the height in the tri-angles Z1, Z2, and Z3 and 0, Z2, and Z3 with thecommon baseline Z2Z3 . In the same way b and -c are ex-plained graphically.

Next we consider the sign of a-, b, and c. Now

30

-0

a

a

0X1-

0

'Z2 (case 1)

zl ,,' a>0, <0,c>0

I Z3

'Z2 (case 2)

,, 5< 0b>0,c<0

Z4

z

'. Z3

Chromatic dispersive power v

Fig. 6. Design example for a four-lens system.

assume that 01/)l > 02/02 > 03/03. When point Z2exists on the upper side (case 1) of the line connectingZ1 and Z3, as shown in Fig. 5, the denominator of Eq.(13) > 0 and the numerator of Eq. (13) > 0, then a >0; likewise b < 0 and > 0. When point Z 2 exists inthe lower side (case 2), the denominator of Eq. (13)< 0 and the numerator of Eq. (13) > 0, then a < 0,and sob > and < 0.

If one can use four lenses made of four differentmaterials, the design of optical systems becomes moreflexible. In this case, one can select two from fourpoints arbitrarily, and the third point is arbitrarilyselected on the line connecting the other two points.For example, as shown in Fig. 6, Z1 andZ4 are selectedat first, then the third point Q is selected on the linethat passes through Z 2 and Z3. Then the designmethod of a triplet is applied.

The thermal expansion of lens support can beintroduced into the athermal chart. The expansionof lens support with length e should be canceled bythe focal-point shift of the optics. In this case, thepoint of athermalization and achromatism is locatedat point [0, -(af/aT)+] instead of the origin as shownin Fig. 7.

C X

w0

Z~~~Z

Z2

0

.

0

_ d Chromatic dispersive power cv

Fig. 7. Design method that includes thermal expansion of the lenssupport.

1 December 1994 / Vol. 33, No. 34 / APPLIED OPTICS 8011

Chromatic dispersive power cv

Fig. 5. Sign of a, b, and c of a triplet.

- l -

_ r

" Z2

II

4. Design Examples of the Multilens System

Here we give design indices through the designexamples with the athermal chart. The multilenssystems were designed in the 8-12-pm range, wherethe available materials are Ge, CdTe, ZnSe, AMTIR-1,GaAs, and ZnS. In Fig. 8(a) chromatic dispersivepower and thermal dispersive power are shown,

15j

Table 1. Effects of Triangle Shape on Lens Power

Material

Type Ge CdTe AMTIR-1 ZnSe

(1) / -2.20 5.11 -1.91(2) hi/4 -1.78 7.33 -4.55

(1) (2)

*Ge

lOCh_

*GaAs- CdTe

AMfIR1 ZnSe .ZnS

I I I

20 40 60Chromatic dispersive power Cv ( x 1O-3)

(a)

Gaes

ZnSe *ZnS

I I I

20 40 60Chromatic dispersive power c& ( x 103)

(b)

20 40 60Chromatic dispersive power c, ( x 103)

(c)Fig. 8. Lenses of materials in the 8-12-pLm wavelength band.

Lenses of materials. (b) Triangles with different shapes,Triangles with different areas.

corresponding to these materials. Below we con-sider two cases relating to the triangles on theathermal chart that satisfy achromatism and ather-malization: one is the shape of the triangle and theother is the area.

A. Triangle Shape

We consider two triangles: One is a triangle withsummits corresponding to Ge, CdTe, and AMTIR-1,and the other is to CdTe, AMTIR-1, and ZnSe asshown in Fig. 8(b). They have the common baselineCdTe AMTIR-1 and have nearly the same area, buttheir shapes are different: The former is flatter thanthe latter. Table 1 shows the calculation results ofthe lens power obtained from the athermal chart.From Table 1, it is found that the flatter trianglerequires higher power.

B. Triangle Area

We consider the triangle (Al) corresponding to Ge,AMTIR-1, and ZnS and make two triangles (A2 , A3 )

as shown in Fig. 8(c): A2 corresponds to Ge, ZnSe,and ZnS and A3 to Ge, AMTIR-1, and ZnSe. In thiscase, AMTIR-1, ZnSe, and ZnS are nearly on the sameline, and so Al can be roughly divided into A2 and A3.In the same way, by the line connecting ZnS andCdTe, Al can be roughly divided into A4 and A5: A4corresponds to Ge, CdTe, and ZnS and A5 to CdTe,AMTIR-1, and ZnS. Table 2 shows the calculationresults of lens power. The area size of the triangle isas follows:

Al> A4 > A2 > A3 > A5-

The results in Table 2 include the effect of not onlyarea but position. However, it is concluded that weshould choose a larger size of triangle.

Table 2. Effects of Triangle Area on Lens Power

Material

Type Ge CdTe AMTIR-1 ZnSe ZnS

Al Pij/+ -0.43 1.78 -0.35A2 i/+ -0.45 2.37 -0.92A3 4,/4 -0.41 2.88 -1.47

A4 $i/4 -0.81 2.06 -0.25

A5 h/4 -2.30 3.76 -0.46

(a)

(c)

8012 APPLIED OPTICS / Vol. 33, No. 34 / 1 December 1994

K

I-iC

50

0 1

- 150

0

* 0

aZ 50

a)

DE

0

0-IX

..00.Co

0

. . . .

. . .

I

3 _

I

5. Conclusion

We have introduced an athermal chart with chro-matic dispersive power and thermal dispersive powerin a Cartesian coordinate and have demonstrated amethod of athermalization and achromatism. Thefeature of this chart is that the power of the lensesthat compose the multilens system is easily obtainedand the system evaluation is easier, and so it iseffective for optical design.

The authors appreciate the guidance and encourage-ment of Fumio Takeda, Okayama University of Sci-ence.

References

1. T. H. Jamieson, "Thermal effects in optical systems," Opt. Eng.20, 156-160 (1981).

2. J. M. Lloyd, Thermal Imaging Systems (Plenum, New York,1975), pp. 257-267.

3. M. Roberts, "Infra-red lenses," U.S. Patent 4,679,891 (14 July1987).

4. P. J. Rogers, "Athermalized FLIR optics," in 1990 Interna-tional Lens Design Conference, G. N. Lawrence, ed., Proc. Soc.Photo-Opt. Instrum. Eng. 1354, 742-751 (1990).

5. P. J. Rogers, "Athermalization of IR optical systems," inInfrared Optical Design and Fabrication, R. Hartmann andW. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng. CR38,69-93 (1990).

6. R. C. Gibbons, "Compact, high cold shield efficiency opticalsystem," U.S. patent 4,431,917 (14 February 1984).

7. L. Rayces and L. Lebich, "Thermal compensation of infraredachromatic objectives with three optical materials," in 1990International Lens Design Conference, G. N. Lawrence, ed.,Proc. Soc. Photo-Opt. Instrum. Eng. 1354, 752-759 (1990).

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