Open Access
Add to BibSonomyAbstract
This paper defines and discusses a glass dispersion formula that is adaptive. The formula exhibits superior convergence with a minimum number of coefficients. Using this formula we rationalize the correction of chromatic aberration per spectrum order. We compare the formula with the Sellmeier and Buchdahl formulas for glasses in the Schott catalogue. The six coefficient adaptive formula is found to be the most accurate with an average maximum index of refraction error of 2.91 × 10−6 within the visible band.
© 2014 Optical Society of America
1. Introduction
A dispersion formula is utilized to relate the refractive index and the wavelength for optical glass. There are several dispersion formulas available and these can be classified into two types. The first type is in the form of an algebraic fraction, and the second type is in a polynomial form. For example, in 1871, Wolfgang Sellmeier developed a formula which belongs to the first type and is [1]
where n is the index of refraction at wavelength λ. Bi and Ci are Sellmeier coefficients, which vary from glass to glass. In 1954, Hans Buchdahl introduced a change of coordinate from wavelength space λ to a chromatic coordinate ω and developed a power series of ω called Buchdahl dispersion formula [2–6]where n is the index of refraction at chromatic coordinate ω, and n0 is the index of refraction at a reference wavelength. Although this model is a polynomial of ω, it is still classified as of the first type model since the relationship between coordinates ω and λ iswhich is in the form of an algebraic fraction. This relationship for ω provides the functional nonlinearity between n and λ in n(λ) space. In Eqs. (2) and (3), λ0 is the reference wavelength; vj and α are Buchdahl coefficients. The coefficient α plays the role of a convergence coefficient which varies with λ0, and lies within narrow limits for common glasses. For instance, if 0.574µm is chosen for λ0 in the visible band, then the optimal value of α is around 2.5. However, in practice the coefficient α is fixed for all glasses to allow the chromatic coordinates to be useful for correcting chromatic aberration [5].A Taylor series, which is an example of the second type of dispersion formula, can be used to describe the dispersive behavior of glass. Such a series is desirable for theoretical studies of chromatic aberration correction. Thus some efforts have been made to develop such a series in wavelength space. For example, a truncated Taylor series centered at the reference wavelength λ0 is [6]
where ap is a Taylor series coefficient. However, a major problem of this polynomial is its weak convergence. A large number of terms are required to achieve a specific accuracy for known index of refraction data. The more terms that are added, the more oscillation (or ripple) can potentially take place. If so, the interpolated index for a given wavelength can have a large error. Thus dispersion formulas that have a small number of terms and that are precise are desirable. However, as shown below the fitting precision can be improved by using more terms.In this paper we present and study an adaptive dispersion formula that provides in the average less index fitting error. The coefficients of the formula when multiplied by the sag of a lens provide directly the amount of primary, secondary, tertiary, etc. spectrum. This decomposition is an insightful way to understand and carry the correction of chromatic aberration in a system of lenses.
2. Adaptive dispersion formula
Our interest in a dispersion formula of the second type is in part motivated by our interest in rationalizing the correction of chromatic aberration. Thus we define a normalized chromatic coordinate as
where Λ is a constant and is the larger value of (λmax-λ0) or (λ0-λmin). Here λmax and λmin are the extreme values of wavelength range of interest. This unitless normalized chromatic coordinate ∆λ has a linear relationship with λ, and its value is limited within ± 1. The central value of ∆λ, which depends on the reference wavelength λ0, is not necessary zero.We now define an adaptive dispersion formula using the normalized chromatic coordinate as
where q is an even positive integer. Aq and K are coefficients that are determined for each specific glass. n0 is the index of refraction at the reference wavelength λ0. Equation (6) is a polynomial on ∆λ and the last term, being odd on ∆λ, is keystone distorted. The formula is adaptive because it selects the best keystone distortion K coefficient for a given glass, and also converges using a small number of coefficients. In practice, as discussed below, for the correction of chromatic aberrations only the A1, A2, A3, A4, A5 coefficients are used.Figure 1 shows graphically how the index of refraction for N-BK7 glass is decomposed with the adaptive dispersion formula.
Fig. 1 Index variation for N-BK7 glass: (a) total variation, (b) linear variation, (c) quadratic variation, (d) cubic variation, (e) quartic variation, and (f) quintic variation.
3. Index fitting
One way to fit measured index of refraction values and obtain formula coefficients for a dispersion formula is by applying the least-squares method [7, 8]. The measured indices are the index data within the visible spectrum listed in the glass catalogue. We performed index fitting within lens design software (Radiant-Zemax) using the built-in orthogonal descent algorithm. A macro language program was written which calculated the RMS of the differences between the calculated indices and the measured indices. Then the dispersion formula coefficients were varied as to minimize the RMS difference and thus the formula coefficients were found. We assume that the eleven values of the measured index of refraction are error free.
4. Dispersion formula fitting accuracy
We select twenty Schott glasses from each zone in the glass map for studying the accuracy of dispersion formulas using λmax = 0.7065188 μm and λmin = 0.4046561 μm. λ0 = 0.546074 μm is chosen for Buchdahl and the adaptive formulas since it can make the optimization converge better. The glasses are F2, LF5, LLF1, N-BAF10, N-BAK4, N-BALF4, N-BASF2, N-BK7, N-FK51A, N-K5, N-KF9, N-KZFS4, N-LAF2, N-LAK9, N-LASF40, N-PK51, N-PSK53A, N-SK16, N-SSK5, and N-SF66. The fitting indices are the measured indices at wavelengths in the visible range: 0.4046561, 0.4358343, 0.4799914, 0.4861327, 0.546074, 0.5875618, 0.5892938, 0.6328, 0.6438469, 0.6562725, and 0.7065188 μm [9].
A comparison between the Sellmeier, Buchdahl, and the adaptive dispersion formula is presented in Table 1. The columns provide the maximum error and RMS error when the dispersion formulas are 1) six coefficient adaptive formula; 2) six coefficient Buchdahl formula; 3) six coefficient Sellmeier formula; 4) four coefficient adaptive formula; and 5) four coefficient Buchdahl formula. One can note that the adaptive formula performs similarly in fitting as the Sellmeier formula. The convergence coefficient K allows the adaptive formula to have small fitting errors. The four coefficient Buchdahl formula (including α as a fitting variable) has larger fitting errors.
Table 1. Fitting Errors for Different Dispersion Formulas (Units: × 10−6)
Table 1 provides in the bottom row a summary of the fitting by providing average values. Clearly, the six coefficient adaptive formula and the four coefficient adaptive formula provide in the average the best index fitting.
When making the comparison using all 119 glasses in the Schott catalogue [9] we found similar results as shown graphically in Fig. 2 and provided in Appendix 1 (Table 2).
Table 2. Index Fitting Comparison (Errors are in units of 10−6)
Moreover, the six coefficient Sellmeier formula for N-BK7 glass is fitted with the four and six coefficient adaptive formulas. This is performed by the orthogonal descent algorithm within the lens design software Radiant-Zemax and a macro written to calculate the RMS of the differences between the indices calculated from Sellmeier and the adaptive formulas. We assume that the 300 values of the indices of refraction calculated from Sellmeier formula are error free. As shown in Fig. 3 the maximum fitting errors of the four and six coefficient adaptive formulas are about 5 × 10−7 and 5 × 10−9 respectively. This indicates that the adaptive formulas can substantially mimic the Sellmeier formula. It also shows that including more terms in a formula can reduce ripple amplitude and improve the fit.
Fig. 3 Index fitting errors of the adaptive formula for N-BK7 as fitted to the six-coefficient Sellmeier formula: (a) six-coefficient adaptive formula, (b) four-coefficient adaptive formula.
5. Correction of chromatic change of focus per spectrum order
The chromatic change of focus ∂λW020 aberration (axial color) for a system of N thin lenses in air is given by [10]
where y is the marginal first-order ray height at the thin lens; c1 and c2 are the curvatures of the surfaces, andis the sag of the lens at height y.Since the chromatic coordinate ∆λ is dimensionless and normalized to unity, the coefficients A1, A2, A3, A4, A5 when multiplied by the lens sag S provide the amount of chromatic aberration as linear, quadratic, cubic, etc. spectrum that are contributed by the lens. This aberration is expressed as an optical path difference between the extreme wavelengths 0.4046561 μm and 0.7065188 μm.
For correcting primary, secondary, and tertiary spectrum, in a system of N thin lenses we must satisfy
Clearly, by expressing the index of refraction as a polynomial on the chromatic coordinate △λ we perform the correction of chromatic aberration per order of spectrum. This is an insightful and useful aberration correction decomposition. Appendix 2 (Table 3) provides the coefficients A1, A2, A3, and K for the four coefficient adaptive formula and for glasses in the Schott catalogue using λ0 = 0.546074 μm, λmax = 0.7065188 μm, and λmin = 0.4046561 μm.Table 3. Coefficients of the Adaptive Dispersion Formula
For N-BK7 glass we have A1 ≅ −0.0082 (see Table 3 in Appendix 2) and for a positive lens with a sag of 1 mm at the edge of the aperture, we have that the contribution to primary spectrum as an optical path difference is −0.0082 mm. This can be corrected with a negative lens made out of F2 glass which has a coefficient A1 ≅ −0.0176 and would require a sag value of ~-0.4688 mm. As another example we mention that the combination of glasses N-FK51A and N-PSK3 allows correcting for primary and secondary spectrum as shown graphically in Fig. 4. This correction is done by cancelation of similar spectrum orders contributed by the positive lens N-FK51A with 1 mm sag at the edge of the aperture and the negative lens N-PSK3 with a sag value of ~-0.6628 mm.
Fig. 4 Cancelation of primary and secondary spectrum in a doublet by using N-FK51A and N-PSK3 glasses: (a) linear term, (b) quadratic term, (c) residual cubic term.
6. Conclusion
We have studied an adaptive dispersion formula that is essentially a truncated polynomial with one adaptive term. A coefficient K in the adaptive term permits the convergence and makes the formula adaptive to each type of glass. We use lens design software and a built-in orthogonal descent optimization algorithm for fitting formula coefficients. We present an index fitting comparison with 119 glasses in the Schott catalogue and for wavelengths in the visible band. As six coefficients are adopted, the adaptive formula provides an average refractive index error of 2.91 × 10−6, while the Buchdahl formula and the Sellmeier formula have in the average refractive index errors of 3.63 × 10−6 and 3.82 × 10−6 respectively. The adaptive formula coefficients when multiplied by the sag of a lens provide directly the amount of chromatic change of focus as primary, secondary, tertiary, etc. spectrums. This is an insightful way to understand and carry out the correction of chromatic change of focus aberration. We also provide a table of coefficients for Schott glasses and for the four coefficient adaptive formula.
Appendix 1
Appendix 2
References and links
1. L. E. Sutton and O. N. Stavroudis, “Fitting refractive index data by least squares,” J. Opt. Soc. Am. 51(8), 901–905 (1961). [CrossRef]
2. P. J. Reardon and R. A. Chipman, “Buchdahl’s glass dispersion coefficients calculated from Schott equation constants,” Appl. Opt. 28(16), 3520–3523 (1989). [CrossRef] [PubMed]
3. G. W. Forbes, “Chromatic coordinates in aberration theory,” J. Opt. Soc. Am. A 1(4), 344–349 (1984). [CrossRef]
4. R. A. Chipman and P. J. Reardon, “Buchdahl’s glass dispersion coefficients calculated in the near infrared,” Appl. Opt. 28(4), 694–698 (1989). [CrossRef] [PubMed]
5. Y. Pi, P. J. Reardon, and D. B. Pollock, “Applying the Buchdahl dispersion model to infrared hybrid refractive-diffractive achromats,” Proc. SPIE 6206, 62062O (2006). [CrossRef]
6. P. N. Robb and R. I. Mercado, “Calculation of refractive indices using Buchdahl’s chromatic coordinate,” Appl. Opt. 22(8), 1198–1215 (1983). [CrossRef] [PubMed]
7. K. S. R. Krishna and A. Sharma, “Evaluation of optical glass composition by optimization methods,” Appl. Opt. 34(25), 5628–5634 (1995). [CrossRef] [PubMed]
8. M. Bolser, “Mercado/Robb/Buchdahl coefficients - an update of 243 common glasses,” Proc. SPIE 4832, 525–533 (2002). [CrossRef]
9. Schott glass catalogue (2011), http://www.us.schott.com.
10. J. Sasian, Introduction to Aberrations in Optical Imaging Systems (Cambridge University, 2013), Chap. 6.
