Apochromatic lenses for near-infrared astronomical instruments
Deqing Ren Abstract. A method based on the Herzberger approach has been in- Jeremy R. Allington-Smith vestigated for the selection of glasses for the apochromatic correction at University of Durham near-infrared (NIR) wavelength. The method avoids the algebraic com- Department of Physics plexity and simplifies the glass selection processes. Doublet and triplet Durham DH1 3LE glass combinations can be chosen directly from the plot of partial disper- United Kingdom sion versus V number. Good combinations of NIR doublets and triplets E-mail: [email protected] are given. Design examples show that the method is practical and effi- cient. © 1999 Society of Photo-Optical Instrumentation Engineers. [S0091-3286(99)02103-0]
Subject terms: optical design; infrared; lenses; optical glass; camera.
Paper 980318 received Aug. 24, 1998; accepted for publication Oct. 21, 1998.
1 Introduction 2 Partial Dispersion and V Number of Glasses In optical design, the first step is to choose optical glasses For NIR glasses, the Sellmeier and Herzberger dispersion to correct chromatic aberration. The control of chromatic formulas are used to calculate the index of refraction. The aberration through the selection of glasses is one of the glass catalogs of various manufacturers, available in digital most extensively studied subjects in the field of lens form, can be used to fit the index data and calculate the design.1–14 Different glass selection methods have been in- coefficients of the dispersion formulas. Then, the dispersion vestigated at visible wavelengths that can yield apochro- formulas can be used to calculate the optical index at any matic and superachromatic lens systems. The term apochro- wavelength. The more data one enters, the more accurate matic and superachromatic mean paraxial color correction will be the fit. The glass index data in this paper are from the infrared glass catalog published with the Zemax optical at three and four wavelengths, respectively. One approach design program.15 These data are provided by Schott16 or uses the Buchdahl glass dispersion equation to select compiled from published sources,17 and their accuracy has glasses for systems of thin lenses that yield apochromatic been confirmed by Albert Feldman’s measurements.18 and superachromatic color correction. Robb and Mercado1 2 For infrared glasses, the Herzberger dispersion formula and Buchdahl use this method to choose two kinds of ma- is often used to calculate the glass index14: terials for a doublet. Another approach uses Herzberger’s relative-partial-dispersion–Abbe-number equation, which can also yield apochromatic and superachromatic color cor- nϭAϩBLϩCL2ϩD2ϩE4ϩF6, ͑1͒ rection. Herzberger and Salzberg14 used this method to se- lect glasses that can yield the same performance. where Lϭ1/(2Ϫ0.028). Schott uses the Herzberger for- However, most of these studies were done at visible mula to calculate its glasses for NIR. The three-term Sell- wavelengths. Little attention has been given to the near- meier dispersion formula can also be used to calculate the 19 infrared ͑NIR͒ wavelengths ͑1to2.5m͒. The use of dou- index of infrared glass with adequate accuracy : blets is often limited by the small number of available glasses and the inability of a doublet to achromatize over K 2 K 2 K 2 the whole NIR wavelength range. We have not found any 2 1 2 3 n Ϫ1ϭ 2 ϩ 2 ϩ 2 . ͑2͒ paper about the choice of glasses for triplets for NIR astro- ϪL1 ϪL2 ϪL3 nomical instruments. The Buchdahl glass dispersion equa- tion is mathematically too complex for the selection of As is done for visible optical wavelengths, for infrared glasses for doublets. From the point of view of optical en- wavelengths the V number and partial dispersion are de- gineering, it is important to find a simple and practical fined for a stated spectral region 1Ͻ2Ͻ3 as method to choose glasses for chromatic-aberration correc- tion. n2Ϫ1 This paper investigates a method that uses Herzberger’s Vϭ , ͑3͒ approach to choose glasses to design doublets and triplets n1Ϫn3 for apochromatic correction in the NIR wavelength range. Doublet glasses or triplet glasses can be chosen directly n2Ϫn3 form the partial-dispersion–V-number plot, without alge- Pϭ , ͑4͒ braic complexity. Finally, two design examples are given. n1Ϫn3
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Table 1 Refractive indices, V number, and partial dispersion of NIR glasses at 20°C. The refractive indices of IRG2, IRG3, IRG7, IRG9, and ZnSe were calculated with the Herzberger formula; the others, with the Sellmeier formula. The silica is fused silica; the ZnSe and ZnS are chemical vapor deposited.
n1 n2 n3 V Glass (1.25 m) (1.65 m) (2.2 m) number P
silica 1.44748 1.44280 1.43501 35.5110 0.6246 LiF 1.38512 1.38190 1.37659 44.7701 0.6217
MgF2-O 1.37214 1.36995 1.36648 65.3879 0.6128 IRG9 1.47745 1.47461 1.47031 66.4437 0.6028 IRG7 1.54755 1.54279 1.53584 46.3737 0.5929
CaF2 1.42746 1.42556 1.42280 91.4696 0.5907 MgO 1.71941 1.71449 1.70762 60.6185 0.5824 IRG11 1.66140 1.65675 1.65060 60.8103 0.5699 Fig. 1 Plot of partial dispersion P versus V number for NIR glasses IRG3 1.81383 1.80709 1.79841 52.3309 0.5627 at wavelengths 1.25, 1.65, and 2.2 mat20°C.
BaF2 1.46719 1.46571 1.46396 144.4753 0.5412 IRG2 1.85741 1.85088 1.84321 59.9108 0.5400
SrTiO3 2.29795 2.28014 2.26082 34.4813 0.5203 ZnS 2.27935 2.26973 2.26338 79.4784 0.3974 NIR astronomical instruments work in the J, H, and K ZnSe 2.46802 2.45278 2.44368 59.7034 0.3739 bands ͑which are also standard atmospheric windows of transparency21͒. In order to avoid refocusing in the three bands, we choose the infrared wavelengths as
1ϭ1.25 m, 2ϭ1.65 m, 3ϭ2.2 m. where n1 , n2 , and n3 are the indices of refraction at the short wavelength 1 , central wavelength 2 , and long wavelength 3 , respectively. It is important to note that Using Eqs. ͑1͒, ͑2͒, ͑3͒, and ͑4͒, we can calculate the indi- 1 , 2 , and 3 are also the wavelengths to be corrected for ces, V number, and partial dispersion P of the most-used chromatic aberration. glasses in the NIR wavelength. The results are given in The plot of partial dispersion versus V number is ex- Table 1. All of these indices are for room temperature tremely important in choosing glasses that will permit sec- ͑20°C͒, and all of these glasses have high transmission. ondary color correction in a lens. Unfortunately, many A plot of V number versus partial dispersion at 20°C is glass catalogs provided with computer design programs shown in Fig. 1. It can be seen that BaF2, ZnS, and ZnSe ͑such as Zemax͒ or by glass companies only provide glasses are far away from other glasses on the plot. This is V-number and P data that are calculated at the F-d-C wave- important in the choice of glass, as will be shown in Secs. lengths. In fact, these data are useless for glass selection, as 3 and 4. As NIR instruments are often operated at cryo- they will be different at visible and NIR wavelengths. genic temperatures, we also calculated the partial disper- Shannon20 has calculated V-number and P values of some sion and V number of all these glasses at 77 K and found visible optical glasses in different wavelength ranges. He that their relative values are almost unchanged. So the plot also compared the plots of V number versus P for these of V number versus partial dispersion at 20°C can be used glasses in the visible F-d-C and the infrared wavelength as a standard map for glass choice at other temperatures. ranges, and found that the glass distributions change dra- matically when the wavelength shifts to the infrared. Herzberger and Salzberg14 calculated the partial disper- 3 Selection of Glass for a Two-Material Doublet sion P of some infrared glasses in the wavelength range 1.5 For simplicity in application, only axial chromatic aberra- to 5.0 m. He used 3.5 m as the central wavelength 2 . tion will be considered. For thin lens doublets in close con- We find that the distributions of glasses on the V-versus-P tact or with a small air space, the conditions for three- 14 plot for differing short wavelength 1 , central wavelength wavelength color correction are 2 , and long wavelength 3 are also dramatically different, and this has generally been ignored. For example if 1 , 2 , ⌽1 ⌽2 and 3 are 1, 1.75, and 2.5 m, respectively, the V-number ϩ ϭ0, ͑5͒ and P of CaF2 glass are 54.6 and 0.488, respectively, com- V1 V2 pared with 91.5 and 0.591 when 1 , 2 , and 3 are 1.25, 1.65, and 2.2 m, respectively. So the V-number and P ⌽ ⌽ data must be calculated according to the wavelengths at 1 2 P1ϩ P2ϭ0, ͑6͒ which one corrects for chromatic aberration. V1 V2
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Table 2 Some possible combinations for two-glass doublets. 4 Selection of Glass for a Three-Material Triplet
Ϫ4 Ϫ4 As we discussed above, for doublets it is required that the Combination ٌl (10 ) Combination ٌl (10 ) two glasses have the same partial dispersion. This limits the possible applications, as only a few glasses have close par- BaF /IRG2 Ϫ0.1419 CaF /IRG9 4.835 2 2 tial dispersion and small residual chromatic aberration. For Ϫ1.900 4.878 BaF2 /SrTiO3 CaF2 /IRG7 a triplet in close contact or with a small airspace, the con-
BaF2 /IRG3 2.333 CaF2 /SiO2 6.058 ditions under which three lenses can be achromatized for three wavelengths are22 BaF2 /IRG11 3.430 CaF2 /LiF 6.638
BaF2 /MgO 4.913 CaF2 /IRG11 Ϫ6.784
BaF2 /IRG7 5.270 CaF2 /IRG3 7.154 ⌽1ϩ⌽2ϩ⌽3ϭ⌽, ͑8͒ BaF2 /SiO2 7.654 MgF2-O/SiO2 6.058
BaF2 /LiF 8.074 MgF2-O/LiF 6.638 BaF /MgF -O 9.053 MgO/IRG7 Ϫ7.371 2 2 ⌽1 ⌽2 ⌽3 ϩ ϩ ϭ0, ͑9͒ CaF2 /MgO Ϫ2.690 MgO/SiO2 Ϫ16.807 V1 V2 V3
⌽1 ⌽2 ⌽3 P1ϩ P2ϩ P3ϭ0. ͑10͒ V1 V2 V3 where ⌽k is the optical power of lens k, ⌽ is the whole power of the two-lens doublet, Vk is the V number of lens k, and Pk is the partial dispersion of lens k. Solving these equations, we get Solving the above equations, we have P1ϭP2 . This means that for two glasses, they must have the same partial dispersion in order to correct three wavelengths. From Table 1 and Fig. 1, it is clear that there are no two glasses 1 ⌽1ϭϪT23 V1⌽, ͑11͒ that have the same partial-dispersion value. Thus, any two- E1 glass doublet will have residual chromatic aberration. The secondary residual chromatic aberration for the two thin 22 lenses can be expressed as 1 ⌽2ϭϪT31 V2⌽, ͑12͒ E2
P1ϪP2 1 lϭϪ . ͑7ٌ͒ V ϪV ⌽ 1 2 1 ⌽3ϭϪT12 V3⌽, ͑13͒ E3 On the P – V-number plot, the slope of a line connecting the two points corresponding to these two glasses is propor- tional to the amount of secondary residual chromatic aber- where ⌽k , Vk , and Pk are the power, V number, and par- ration. The amount of excess power that must be introduced tial dispersion of lens k, and in the element of the lens in using these glasses to achieve secondary color correction is inversely proportional to the length of the line separating the two glass points. It is no- ticeable that in Fig. 1, along the axis of partial dispersion P, 1 E1ϭϪ ͓V1͑P2ϪP3͒ϩV2͑P3ϪP1͒ ZnS and ZnSe are far away from other glasses, and this V2ϪV3 means that ZnS and ZnSe are not suitable for forming dou- ϩV3͑P1ϪP2͔͒, blets with other glasses. BaF2 and CaF2 are far away from other glasses in the V-number direction and are very close in partial dispersion to some other glasses. This means that BaF2 and CaF2 can be used to form doublets with other 1 E2ϭϪ ͓V1͑P2ϪP3͒ϩV2͑P3ϪP1͒ glasses. BaF2 is better than CaF2 because of its larger V V3ϪV1 23 number. Oliva and Gennari confirmed that BaF2 /IRG2 ϩV3͑P1ϪP2͔͒, has better performance than CaF2 /IRG7. Table 2 lists some possible combinations for doublets and the residual chromatic aberration. The residual aberra- tion is calculated per unit ⌽. The smaller the residual chro- 1 E ϭϪ ͓V ͑P ϪP ͒ϩV ͑P ϪP ͒ matic aberration, the better the two-lens doublet corrects 3 V ϪV 1 2 3 2 3 1 chromatic aberration. In practical application, the two 1 2 lenses can be separated slightly to facilitate cooling. ϩV3͑P1ϪP2͔͒,
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eral, a good triplet will have a large triangle area corre- sponding to the three glasses and will have small power differences among the three elements. The basic glass selection problem is to obtain the mini- mum values of the optical powers ⌽1 , ⌽2 , ⌽3 , since the smaller the optical power, the smaller the higher-order ab- errations. Some possible combinations for a triplet are listed in Table 3. All of the triplets in Table 3 have a rea- sonable power distribution except CaF2 /BaF2 /ZnSe, which has a larger power in its first and second elements and is listed just for comparison. The power of each thin lens is calculated with that of the triplet as the unit. The square root of the averaged and squared weighted refractive power ⌽k of all the lens elements can be used to evaluate the performances.24 This parameter has the advantage of being independent of lens scaling, aperture size, and field angle. It Fig. 2 P – V-number plot of the glasses used in the three-lens triplet. is termed W, and for a triplet it can be expressed as
P1ϪP2 P2ϪP3 P3ϪP1 T12ϭ , T23ϭ , T31ϭ . 1/2 1/2 V ϪV V ϪV V ϪV 1 2 1 2 2 2 1 2 2 3 3 1 Wϭ ⌽ ϭ ͑⌽ ϩ⌽ ϩ⌽ ͒ , ͑14͒ ͩ N ͚ k ͪ ͫ3 1 2 3 ͬ
The meanings of E1 , E2 , E3 , T12 , T23 , and T31 can be seen on a P – V-number plot. For example, for a silica/BaF2 /ZnSe triplet, glass 1 is fused silica, glass 2 is BaF2, and glass 3 is ZnSe. If we plot the three chosen where N is the number of the elements in the triplet. The glasses on the P – V-number plot shown in Fig. 2 and then triplets are listed according to their W values in Table 3. The maximum paraxial rms spot radius (R ) is also join the three points to form a triangle, E1 is the vertical max distance of glass 1 to the line joining the other two glasses listed in Table 3. The paraxial rms spot radii were calcu- and is negative if glass 1 falls below the line. Similarly, E2 lated at the three wavelengths 1 , 2 , and 3 by using the is the vertical distance of glass 2 to the line joining the Zemax optical design program, allowing correction of other two glasses, and E3 is the vertical distance of glass 3 spherical aberration and apochromatism. For the calcula- tion, all the triplets have focal length 100 mm at f/5 with to the line joining the other two. Only E3 is shown in Fig. the thickness of 5 mm for all the three elements. It is ob- 2. The ratio Tij is the slope of the line joining glass i and glass j. All three lenses will be as weak as possible if we vious that except for CaF2 /BaF2 /ZnSe, all the triplets have select glass types having large E1 , E2 , E3 and small T12 , a similar W, so they all have a similar Rmax value. The T23 , T31 . If two glasses have the same or close V-number CaF2 /BaF2 /ZnSe triplet has a larger Rmax than the other values, the slope of the line will be infinite or large, and this triplets, because two of its elements (CaF2 and BaF2) have situation must be avoided when we choose glasses. In gen- larger optical power.
Table 3 Some possible combinations for three-glass triplets.
Combination 1 2 3 WRmax (m)
SrTiO3 /BaF2 /ZnSe Ϫ0.342936 1.228050 0.114886 0.739 0.792
SrTiO3 /BaF2 /ZnS Ϫ0.346881 1.271849 0.075034 0.762 0.785
silica/BaF2 /ZnSe Ϫ0.234825 1.431636 Ϫ0.196811 0.846 0.801
LiF/BaF2 /ZnS Ϫ0.317755 1.521649 Ϫ0.203894 0.905 0.765
IRG7/BaF2 /ZnSe Ϫ0.373084 1.521516 Ϫ0.148432 0.908 0.755
silica/BaF2 /ZnS Ϫ0.242130 1.556429 Ϫ0.314299 0.926 0.763
IRG3/BaF2 /ZnSe Ϫ0.507877 1.582340 Ϫ0.074463 0.960 0.807
IRG7/BaF2 /ZnS Ϫ0.381774 1.617015 Ϫ0.235241 0.969 0.743
IRG3/BaF2 /ZnS Ϫ0.509999 1.618521 Ϫ0.108523 0.981 0.754
CaF2 /BaF2 /ZnSe Ϫ1.17137 2.397588 Ϫ0.226217 1.546 1.815
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silica/BaF2 /ZnSe, and the doublet is silica/BaF2. The last element in the two cameras is a ZnSe field lens. All the optical specifications for the two cameras are the same. The
only difference is that the silica/BaF2 doublet is replaced by a silica/BaF2 /ZnSe triplet in the triplet-triplet camera. Al- though some glass combinations have better performance, we only choose classical glasses in view of their good physical characteristics. Some glasses, such as IRG2, IRG3, IRG7, IRG9, and IRG11, are avoided, as Schott will most probably stop the production of these little-used
glasses. After silica/BaF2 /ZnSe is chosen, the doublet silica/BaF2 is chosen in order to minimize the number of glasses. The design is optimized at the cryogenic tempera- ture of 70 K. Although the two cameras are only optimized at the wavelengths 1.25, 1.65, and 2.2 m, the chromatic aberra- tion is also well corrected at other wavelengths between 1 and 2.5 m. Figure 4 shows spot diagrams of the two cam- eras. It is not surprising that the triplet-triplet camera has Fig. 3 Optical layout of f/2 NIR camera. All lens surfaces are spheri- better optical performance than the triplet-doublet camera. cal. Spot diagrams are displayed in Fig. 4. This is because the silica/BaF2 /ZnSe triplet is better than the silica/BaF2 doublet for color and other aberration cor- rections. Once the glasses have been chosen, the lenses can be assembled in any order. Again, in practical application, the component lenses of the triplet can be separated slightly to facilitate use in a cryogenic instrument. 6 Conclusions The choice of a doublet is limited by the fact that the two glasses must have the same partial dispersion. In principle, 5 Design Examples a doublet has residual chromatic aberration when the partial dispersions of the two glasses are not equal. For triplets, we Using glasses from Tables 2 and 3, two camera designs for have more freedom to choose glasses for apochromatic lens NIR spectrograph have been optimized to illustrate color design. We have developed a simple procedure to identify correction at 1.25, 1.65, and 2.2 m. Both cameras have suitable combinations of glasses for use in NIR and have focal length 150 mm at f/2, and the aperture stops are one focal length ͑150 mm͒ before the first lens surface. The shown that some classical glasses, such as fused silica, collimated beam sizes are 75 mm. The detector array is BaF2, ZnSe, and LiF, can be used to form a high- assumed to be a Rockwell 1024ϫ1024 with 18.5-m pixel performance triplet. The design examples show that a size. The optical layouts are shown in Fig. 3. One camera triplet-doublet or triplet-triplet combination can be used to comprises a detached triplet and a detached doublet, and design a faster camera, and the triplet has better perfor- the other comprises two detached triplets. The triplet is mance than the doublet for apochromatic lens design.
Fig. 4 Spot diagrams at several wavelengths and field positions (expressed in pixel units with respect to the center of the detector array). The box size is 20 m, and the pixel size is 18.5 m.
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Acknowledgment 21. S. E. Persson and S. C. West, ‘‘A near-infrared camera for Las Cam- panas observatory,’’ Publ. Astron. Soc. Pac. 104, 204 ͑1992͒. We wish to thank Dr. B. Rauscher for his helpful com- 22. R. Kingslake, Lens Design Fundamentals, pp. 81–85, Academic Press ments. ͑1978͒. 23. E. Oliva and S. Gennari, ‘‘Achromatic lens systems for near infrared References instrument,’’ Astron. Astrophys. Suppl. Ser. 114, 179 ͑1995͒. 24. J. M. Sasian and M. R. Descour, ‘‘Power distribution and symmetry 1. P. N. Robb and R. I. Mercado, ‘‘Glass selection for hyperachromatic in lens systems,’’ Opt. Eng. 37͑3͒, 1001–1004 ͑1998͒. triplets,’’ J. Opt. Soc. Am. 73, 1882 ͑1983͒. 2. H. A. Buchdahl, ‘‘Many-color correction of thin doublets,’’ Appl. Opt. 22, 1878 ͑1985͒. 3. N. v. d. W. Lessing, ‘‘Selection of optical glasses in apochromats,’’ J. Deqing Ren received BS and MS degrees Opt. Soc. Am. 47, 955 ͑1957͒. from East China Institute of Technology, 4. N. v. d. W. Lessing, ‘‘Further consideration on the selection of optical China, in 1985 and 1988, respectively. He glasses in apochromats,’’ J. Opt. Soc. Am. 48, 269 ͑1958͒. was with the Kunming Institute of Technol- 5. N. v. d. W. Lessing, ‘‘Selection of optical glasses in superachro- mats,’’ Appl. Opt. 9, 1955 ͑1970͒. ogy from 1988 to 1996, researching on in- 6. R. E. Stephens, ‘‘Selection of glasses for three-colour achromats,’’ J. frared optical and system design. From Opt. Soc. Am. 49, 398 ͑1959͒. 1996 to 1997, he was a visiting scholar of 7. R. E. Stephens, ‘‘Four-colour achromats and superachromatas,’’ J. the astronomical instrumentation group at Opt. Soc. Am. 50, 1016 ͑1960͒. University of Durham, England. He is cur- 8. R. I. Mercado and P. N. Robb, ‘‘Design of thick doublets corrected at rently studying for a PhD degree at Univer- four and five wavelengths,’’ J. Opt. Soc. Am. 71, 1639 ͑1981͒. sity of Durham and involved in the design 9. P. J. Sands, ‘‘Inhomogeneous lenses, chromatic aberrations,’’ J. Opt. of optical and infrared camera for the FMOS spectrograph for the 8 Soc. Am. 61, 777 ͑1971͒. 10. G. W. Forbes, ‘‘Chromatic coordinates in aberration theory,’’ J. Opt. meter SUBARU Telescope. He has published over 10 journal pa- Soc. Am. A 1, 344 ͑1984͒. pers. His research interest includes optical design and astronomical 11. P. N. Robb and R. I. Mercado, ‘‘Calculation of refractive indices instruments. using Buchdahl’s chromatic coordinate,’’ Appl. Opt. 22, 1198 ͑1983͒. 12. P. N. Robb, ‘‘Selection of optical glasses. 1: Two materials,’’ Appl. Opt. 24, 1864 ͑1985͒. Jeremy R. Allington-Smith obtained a 13. R. D. Sigler, ‘‘Glass selection for airspaced apochromats using the PhD in astronomy from Cambridge Univer- Buchdahl dispersion equation,’’ Appl. Opt. 23, 4311 ͑1986͒. sity. Subsequently, he worked at the Mull- 14. M. Herzberger and C. D. Salzberg, ‘‘Refractive indices of infrared ard Space Science Laboratory of Univer- optical material and color correction of infrared lenses,’’ J. Opt. Soc. sity College, London, before joining the Am. 52, 420 ͑1962͒. staff at Durham University, England. In ad- 15. ZEMAX Optical Design Program, User’s Guide, Version 7.0 ͑1998͒. 16. Schott Spec Sheet ͑1991͒. dition to research in astrophysics, he is 17. Handbook of Optics, Vol. II, McGraw-Hill ͑1995͒. working on a variety of integral field and 18. A. Feldman, D. Horowitz, R. M. Waxler, and M. J. Dodge, Optical multiobject spectrographs including those Materials Characterization, Technical Note 993, U.S. National Bu- for the GEMINI Telescopes Project which reau of Standards ͑1979͒. is building 8m telescopes in Hawaii and 19. L. E. Sutton and O. N. Stavroudis, ‘‘Fitting refractive index data by Chile. least squares,’’ J. Opt. Soc. Am. 51, 901–905 ͑1961͒. 20. R. R. Shannon, ‘‘Spectral plots for optical glass selection,’’ Opt. Eng. 35͑10͒, 2995–3000 ͑1996͒.
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