Method of glass selection for color correction in optical system design

Bráulio Fonseca Carneiro de Albuquerque; Jose Sasian; Fabiano Luis de Sousa; Amauri Silva Montes

doi:10.1364/OE.20.013592

1. Introduction

Multi-spectral imaging instruments are widely used in scientific instrumentation. The different spectral bands, which frequently go beyond the visible region of spectrum, can allow extraction of many important features and chemical-physical information from the objects been imaged.

Common examples of this kind of instrument are multispectral satellite remote sensing cameras, multispectral microscopes, and astronomical telescope multispectral cameras. These instruments sometimes cover spectral bands from the UV to the thermal IR. Depending on the instrument specification, an all-reflective solution for the optical system, which would be chromatic aberration-free, is not always feasible, due to some disadvantages of this kind of solution [1]. The design of a refractive optical system that can cover a wide spectral band providing good image quality is not an easy task. According to Rayces and Aguilar [2], two barriers impose limitations on an optical system performance, light diffraction and chromatic aberration.

The chromatic aberration in imaging forming optical systems is a well-known issue studied since the XVII century. As pointed out by Sigler [3], this topic has been one of most investigated in optical design.

Several graphical and mathematical methods for the selection of optimum glass combinations for the correction of chromatic aberration have been proposed [2–13]. However, the problem of glass selection is wide in scope and in our opinion is not yet completely solved. Even a recent publication about optimum glass selection [14] brings no relevant contributions for the subject in our opinion.

Some contemporary methods propose the use of evolutionary [15,16] and hybrid [17] optimization algorithms for the optimal glass selection. Despite of reporting excellent results, we believe these techniques can be computationally demanding due to the extremely huge number of different possibilities available even for a reasonable simple optical system. Moreover, these methods do not guaranty that the best set of glasses has been found.

Fischer et al [18] mention that the glass selection in optical design has a mystique and tends to be both a science and an art. Our goal in this paper is to present a synthesis method that systematizes the task of glass selection for the design of color corrected optical systems, making this an objective task. The proposed method is based on the unification of two other methods proposed in the literature [2,7] with some important contributions added, and a multi-objective approach for the problem.

In the next Section we present the motivation that conducted us to the development of this new method of glass selection. In the Section 3 we present the background for the proposed method, in Section 4 we explain the proposed method itself. In Section 5 we present one example of the application of the method. Concluding this work in Section 6.

2. Motivation

The studies presented herein were motivated by the request for the feasibility study of a single refractive optical system capable of covering and providing excellent image quality in five spectral bands, going from the blue (0.45-0.485-0.52 μm), passing through green (0.52-55-0.59 μm), red (0.63-0.66-0.69 μm), NIR (0.77-0.83-0.89 μm) and reaching the SWIR (1.5-1.6-1.7 μm) spectral region.

Optical system covering wide band with good image quality is a challenge due to the chromatic aberration. In this case a detailed study of how to design a broadband system like this had to be conducted. The success of such system lies on the selection of the right set of optical glasses used in the design [2,7,9,11,16]. For this purpose we started to study the available methods of glass selection in the literature. During this survey, we identified points that could be better developed in the available methods and we ended up with an improved technique that we present in this paper.

Using the developed method it was possible to design a preliminary system for the five spectral bands camera, which complies with the main requirements imposed to the system, as we show in Section 5.

3. Background of the proposed method

After studding many methods of glass selection available in the literature [2–13], we realized that the one proposed by Mercado and Robb [7] is the most theoretically rigorous and general. The Mercado-Robb method considers in the formulation different number of glasses used in the set, as well as different number of wavelengths for which the minimization of chromatic aberration is desired. It is possible to affirm that other methods presented in the literature [e.g 4–6,9,11,13.] can be seen as special cases of this method.

Despite the general formulation of the Mercado-Robb method and the excellent discussion provided in reference [7], the authors solve the problem for practical purposes just in some specific cases. The cases for two glasses corrected from 2 to n wavelengths are very well presented with all the necessary details. Nevertheless, for more than two glasses, only some particular cases are discussed. One reason is that for more than two glasses, the metric adopted to define how good a specific set of glasses is at color correction for n specific wavelengths, becomes difficult to define and interpret geometrically. Furthermore, the adopted method to calculate the optical power of each glass type does not have a general equation, also becoming complicated in these cases. The method presented in this paper improves the Mercado-Robb method with contributions that address some of the practical implementations issues.

In spite of the method of color correction proposed by Rayces and Aguilar [2] being limited to two glasses and three wavelengths, it establishes and makes use of some metrics that appear not to have been reported before in glass selection theory. These metrics are not related to color correction but are important for verifying if a set (in their case a pair) of glasses can provide a successful design. In contrast to other glass selection methods, the Rayces-Aguilar method uses as input not only the wavelengths, but also the focal length and the numerical aperture of the designed system. In the method presented in this paper, we incorporate this metrics proposed by Rayces and Aguilar [2].

With different metrics of dissimilar physical natures for each possible glass arrangement, the use of a multi-objective approach was very convenient and helpful to filter out the non-dominated solutions and organize them in different Pareto rankings, helping the selection of the most appropriate glass combination solution for the problem.

3.1-The Mercado and Robb method with some new contributions

The index of refraction of optical materials is a function of the wavelength. Several mathematical models have been proposed to describe this dependence. Some are based in physical models other are simply empirical functions [19]. One of these models, proposed by Buchdahl [20] is given by Eq. (1).

N (λ) = N_{0} + ν_{1} ω (λ) + ν_{2} ω {(λ)}^{2} + \dots + ν_{n} ω {(λ)}^{n}

This model, as many others, is based on a Taylor series. N represents the refraction index for wavelength λ. N₀ is the refraction index in a reference wavelength λ₀, and ω is a function of the wavelength λ that is called chromatic coordinate:

ω = \frac{δ λ}{1 + α δ λ}

where δλ = λ-λ₀, and α is a universal constant taken as 2.5 [7]. The dispersion coefficients ν_n, are particular to a given glass. This dispersion equation proposed by Buchdahl converges rapidly and can model optical glasses to a very good accuracy using only a few terms in the series [20].

The develop method of glass selection for color correction takes advantage of this dispersion equation. If a set of glasses is needed to minimize the chromatic aberration for n wavelengths, Eq. (1) is expanded to include up to the n-1th algebraic power term. Then a system of linear equations is obtained to compute the dispersion coefficients ν_n of each glass were the number of unknowns is equal to the number of equations.

By passing N₀ to the left side of Eq. (1) and by dividing both sides by the constant N₀ –1, we obtain:

D (λ) = \sum_{i = 1}^{n - 1} η_{i} ω {(λ)}^{i}

where: _{$D (λ) = δ N (λ) / (N_{0} - 1)$} ; _{$δ N (λ) = N (λ) - N_{0}$} and _{$η_{i} = ν_{i} / (N_{0} - 1)$}. The term D(λ) is called dispersive power.This equation is very important since the method presented in [7] is mainly based on it.

The optical power ϕ of a lens is defined as the inverse of the it’s focal length f:

ϕ = \frac{1}{f}

The optical power of a single thin lens for a wave λ is given by the relationship:

ϕ (λ) = [N (λ) - 1] (C_{1} - C_{2})

where C₁ and C₂ are the lens curvature.

For a specific optical material and a defined optical power, the quantity (C₁ – C₂) must be a constant, conveniently called K. Thus we can write:

ϕ (λ) = [N (λ) - 1] K

As a consequence, the power of a thin lens at λ₀ can be expressed by:

ϕ (λ_{0}) = [N (λ_{0}) - 1] K

By use of_{$D (λ) = δ N (λ) / (N_{0} - 1)$}, and _{$δ N (λ) = N (λ) - N_{0}$}, together with Eq. (7), we can write for the optical power:

ϕ (λ) = ϕ (λ_{0}) [1 + D (λ)]

For a system of k thin lenses in contact, the resulting optical power for the reference wavelength λ₀ is computed by:

Φ (λ_{0}) = \sum_{j = 1}^{k} ϕ_{j} (λ_{0})

Using Eq. (8) and Eq. (9), the total optical power for any wavelength λ can be written as:

Φ (λ) = Φ (λ_{0}) + \sum_{j = 1}^{k} ϕ_{j} (λ_{0}) D_{j} (λ)

Assuming that each one of the k lenses is made out of a different glass, where k≥2, the mathematical conditions for having an achromatized optical system in n wavelengths, where n≥2, can be given by:

\begin{matrix} Φ (λ_{1}) = Φ (λ 2) \\ Φ (λ_{2}) = Φ (λ 3) \\ ⋮ \\ Φ (λ_{n - 1}) = Φ (λ n) \end{matrix}

Using Eq. (10), Φ(λ₁), Φ(λ₂), Φ(λ₃), ..., Φ(λ_n) can be transcribed in the following form:

\begin{matrix} Φ (λ_{1}) = Φ (λ_{0}) + ϕ_{1} (λ_{0}) D_{1} (λ_{1}) + \dots ϕ {}_{k}{(λ_{0})} D_{k} (λ_{1}) \\ Φ (λ_{2}) = Φ (λ_{0}) + ϕ_{1} (λ_{0}) D_{1} (λ_{2}) + \dots ϕ {}_{k}{(λ_{0})} D_{k} (λ_{2}) \\ ⋮ \\ Φ (λ_{n}) = Φ (λ_{0}) + ϕ_{1} (λ_{0}) D_{1} (λ_{n}) + \dots ϕ {}_{k}{(λ_{0})} D_{k} (λ_{n}) \end{matrix}

Using the set of Eq. (12), the conditions for achromatized optical systems are:

\begin{matrix} ϕ_{1} (λ_{0}) \cdot (D_{1} (λ_{1}) - D_{1} (λ_{2})) + ... ϕ_{k} (λ_{0}) \cdot (D_{k} (λ_{1}) - D_{k} (λ_{2})) = 0 \\ ϕ_{1} (λ_{0}) \cdot (D_{1} (λ_{2}) - D_{1} (λ_{3})) + ... ϕ_{k} (λ_{0}) \cdot (D_{k} (λ_{2}) - D_{k} (λ_{3})) = 0 \\ ⋮ \\ ϕ_{1} (λ_{0}) \cdot (D_{1} (λ_{n - 1}) - D_{1} (λ_{n})) + ... ϕ_{k} (λ_{0}) \cdot (D_{k} (λ_{n - 1}) - D_{k} (λ_{n})) = 0 \end{matrix}

The difference in the dispersive power of a particular glass j over the wavelength range λ₁< λ< λ₂, can be written in simplified form as:

D_{j} (λ_{1}, λ_{2}) = D_{j} (λ_{1}) - D_{j} (λ_{2})

By using the dispersive power definition in Eq. (3), we can rewrite Eq. (14) in terms of the chromatic coordinate ω as:

D_{j} (λ_{1}, λ_{2}) = \sum_{i = 1}^{n - 1} η_{i j} [ω^{i} (λ_{1}) - ω^{i} (λ_{2})]

Thus we can write the conditions for achromatized optical systems, expressed in Eq. (13), in matrix form as:

Δ \bar{Ω} \cdot \bar{η} \cdot \bar{Φ} = \bar{0}

where, _{$Δ \bar{Ω}$} is a square matrix of order n-1 x n-1:

Δ \bar{Ω} = [\begin{matrix} (ω_{1} - ω_{2}) & (ω_{1}^{2} - ω_{2}^{2}) & \dots & (ω_{1}^{n - 1} - ω_{2}^{n - 1}) \\ (ω_{2} - ω_{3}) & (ω_{2}^{2} - ω_{3}^{2}) & \dots & (ω_{2}^{n - 1} - ω_{3}^{n - 1}) \\ \dots \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ (ω_{n - 1} - ω_{n}) & (ω_{n - 1}^{2} - ω_{n}^{2}) & \dots & (ω_{n - 1}^{n - 1} - ω_{n}^{n - 1}) \end{matrix}]

_{_$\bar{η}$} is a matrix of order n-1 x k:

\bar{η} = [\begin{matrix} η_{11} & η_{12} & \dots & η_{1 k} \\ η_{21} & η_{22} & \dots & η_{2 k} \\ η_{31} & η_{32} & \dots & η_{3 k} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ η_{(n - 1) 1} & η_{(n - 1) 2} & \dots & η_{(n - 1) k} \end{matrix}]

_{_$\bar{Φ}$} is a matrix of order k x1:

\bar{Φ} = [\begin{matrix} ϕ_{1} (λ_{0}) \\ ϕ_{2} (λ_{0}) \\ ⋮ \\ ϕ_{K} (λ_{0}) \end{matrix}]

and _$\bar{0}$ is a matrix of the n-1 x 1 order:

\bar{0} = [\begin{matrix} 0 \\ 0 \\ 0 \\ ⋮ \\ 0 \end{matrix}]

The matrix _{$Δ \bar{Ω}$} is a square and doubtless nonsingular. As a consequence its inverse _{$(Δ {\bar{Ω}}^{- 1})$}exists. Multiplying both sides of Eq. (16) by _{$Δ {\bar{Ω}}^{- 1}$} results in a condition to obtain a solution free from chromatic aberration for all the wavelengths defined:

\bar{η} \cdot \bar{Φ} = \bar{0}

Equation (21) has a nontrivial solution (i.e. _{$\bar{Φ} \neq \bar{0}$}) if and only if the matrix _$\bar{η}$ rank is lower than k (i.e. not a full rank matrix). This happens when there is a perfectly linear dependence among the columns of matrix _$\bar{η}$. Nevertheless, for any practical and meaningful situation, where k≤n-1, the linear dependence will virtually never be mathematically exact. As a consequence, the matrix _$\bar{η}$ rank will always be equal to k. This result makes the rank of matrix _$\bar{η}$ an inefficient metric either to identify sets of glasses that are free from chromatic aberration in the defined wavelengths, or to compare the residual chromatic aberration among the different possible combination of glasses.

To solve this problem, Mercado and Robb provide a geometrical interpretation of the Eq. (21). In this way, they suggest a geometric metric to verify how good a set of glass is at color correction for a given set of wavelengths. The metric is easy to understand and visualize for the case of two glasses. Nevertheless, for more than two glasses the interpretation changes and becomes complicated. Another drawback is that the metric has no physical meaning.

In this paper we propose a different metric to verify how good a specific set of glasses is at minimizing the chromatic aberration for a given set of wavelengths. The proposed metric has a general form, not depending on the number of glasses used in the combination, and has a direct physical meaning. This new metric is presented and explained in some paragraphs ahead.

To minimize or correct the chromatic aberration, not only a specific set of compatible glasses must be selected, but also the right optical power for the lenses made with each one of these materials must be used. To calculate the optimum power of each glass that minimizes the chromatic aberration, both Eq. (16) and Eq. (9) are used. To simplify the computation, we normalize the focal length of the optical lens system for λ₀. As a consequence Eq. (9) becomes:

\sum_{j = 1}^{k} ϕ_{j} (λ_{0}) = 1

This equation can be written in a matrix form as:

\bar{S} \cdot \bar{Φ} = 1

where _$\bar{S}$ is a row vector of order 1xk, with all elements equal to one.

Putting together Eq. (16) and Eq. (22) we obtain:

[\begin{matrix} \bar{S} \\ Δ \bar{Ω} \cdot \bar{η} \end{matrix}] \cdot \bar{Φ} = \hat{e}

where _$\hat{e}$ is a column vector of order nx1 with the first element equal to one and the others zero as shown below:

\hat{e} = [\begin{matrix} 1 \\ 0 \\ 0 \\ ⋮ \\ 0 \end{matrix}]

Defining _{$\bar{G} = [\begin{matrix} \bar{S} \\ Δ \bar{Ω} \cdot \bar{η} \end{matrix}]$}, and assuming that n ≥ k and k≥2, we estimate an optimum _$\bar{Φ}$ applying in Eq. (24) the least square method, what results the following equation:

\hat{\bar{Φ}} = {({\bar{G}}^{t} \cdot \bar{G})}^{- 1} \cdot {\bar{G}}^{t} \cdot \hat{e}

Equation (26) computes the optimum power of the lenses made with each one of the glasses considered in the set that minimizes the square sum of the chromatic change of power for the n defined wavelengths. We point out that the equations provided by Mercado and Robb, to compute the optical powers, are only related to some specific situations and do not use all the glass information available. In contrast, the equation presented herein is general, uses all the glass dispersion coefficients available, and provides the minimum chromatic aberrations for the glass set considered, in the n given wavelengths.

Now, it is possible to use the vector _{$\hat{\bar{Φ}}$} in Eq. (16), to obtain the minimum chromatic change of power _{$\bar{C C P}$}, as expressed in Eq. (27).

\bar{C C P} = Δ \bar{Ω} \cdot \bar{η} \cdot \hat{\bar{Φ}}

Our metric to verify how suitable a specific set of glasses is for minimizing chromatic aberration, for a given set of n wavelengths, is now established as the modulus of the vector _{$\bar{C C P}$}.

As we have normalized the optical power (see Eq. (22)), we obtain an excellent approximation for the chromatic focal shift by multiplying the vector _{$\bar{C C P}$} by the desired effective focal length F for the optical system.

[\begin{matrix} f (λ_{2}) - f (λ_{1}) \\ f (λ_{3}) - f (λ_{2}) \\ ⋮ \\ f (λ_{n}) - f (λ_{n - 1}) \end{matrix}] \approx \bar{C C P} \cdot F

As the chromatic focal shift is proportional to _{$\bar{C C P}$} it clearly gives physical meaning to our metric.

3.2-The Rayces-Aguilar method

Rayces and Aguilar [2] proposed a method of glass selection where not only the chromatic correction is considered, but also aberrations, that according to the authors, cannot be corrected, namely spherochromatism and fifth order spherical aberration.

The Rayces-Aguilar method is based on an exhaustive search of combination of pairs of glasses. The possible arrangements of glasses, deriving from a glass catalog, are tested. For each glass set possibility the power of the glasses are computed to produce a thin achromatic doublet solution for the two extreme wavelengths considered. The chromatic aberration for the middle wavelength, also called the secondary spectrum, is computed. Based on the power of the elements of the doublet and on the desirable aperture of the system, a first weeding out of potentially useless solutions is carried out. This eliminates solutions with steep curves, what is an indication of high-order monochromatic aberrations, which are difficult to correct or balance. In the next step, the radius of each surface is computed to produce an aplanatic solution to third-order approximation using structural aberration coefficients. Paraxial rays are then traced to compute third–order sphero-chromatism and fifth-order spherical aberration. Based on the magnitude of these aberrations, a second glass arrangement elimination is carried out.

The output of the Rayces-Aguilar method is a table with solutions that comply with the limits imposed for each aberration, ranked according to the secondary spectrum value. The method provides a certain level of confidence for glass combinations solutions that may provide a successful design.

3.3-The multi-objective approach

Despite being frequently considered as mono-objective, practical optimization problems have in general more than one objective or criteria, which usually are conflicting. In the problem of finding the best glass combinations for color correction we pursue more than one objective and therefore the use of a multi-objective approach [21] is appropriate.

When a problem is treated as multi-objective, usually there is a set of solutions, and not only one solution. One solution in this set cannot be considered, in principle, better than another solution in the same set, because at least it will be worse than another solution in one aspect or objective. This set of solutions is known as non-dominated solutions. When these solutions are plotted in the objective-functions space they form the thus called Pareto front.

To illustrate the ideas of dominance, non-dominance and Pareto front consider Fig. 1 where blue and red dots represent solutions of a multi-objective problem as plotted in the objective-function space F₁ and F₂.

Fig. 1 The graph shows solutions for a generic min-min multi-objective problem, plotted in the objective-functions space F₁ and F₂. Dominated solutions are represented in blue, while red dots represent non-dominated solutions.

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Objective functions are metrics used to evaluate a specific characteristic of a solution. The example considered in Fig. 1 represents a two objective minimum-minimum problem. This means that the smaller the values of F₁ and F₂ the better the solution is.

The red dots represent the non-dominated solutions while the blue dots represent the dominated solutions. A given solution “A” is considered dominated when there is a solution “B” with at least one of objectives better than the objectives at solution “A”. The dominance relationship among the dominated solutions can also be considered; for example, the solution 1 dominates solutions 2 and 3 [21].

When the non-dominated solutions are plotted on the objective function space, they form the so-called Pareto front, represented in the Fig. 1 by the dashed line connecting the red dots.

Another useful concept used in this paper is the Pareto rank [22]. For a set of possible solutions of a specific problem the dominance definition can be applied several times. Each time the previous non-dominated solutions are removed, giving place to the formation of a new Pareto front. The different Pareto fronts that result are classified by ranking. For example, in Fig. 1 the red dots forming the Pareto ranking equal to 1. Solutions 5, 6, 1 and 4 are the ones forming the Pareto ranking 2 and so on.

The dominance, non-dominance, Pareto front and Pareto rank concepts can be used for multi-objective problems containing any number of objectives.

4. The synthesis method of glass selection

With the background presented in the last section, the explanation of our method of glass selection becomes straightforward. Its implementation involves several steps.

Step 1. As input data for the method, the designer must provide the effective focal length F, the f number F/#, the n wavelengths that covers the desired spectral range, and the number of the primary wavelength λ₀. A glass catalog and the number of glasses used in the combination (i.e. 2, 3, 4, etc) must also be specified.

Step 2. At the outset, the first n-1 dispersion coefficients η_i, are calculated for each glass in the catalog. For that, the n specified wavelengths and their respective refractive index in the corresponding glass are used in Eq. (3). This results in a system of linear equations with n-1 equations and n-1 unknowns, that when solved provides the η_i dispersion coefficients. With the specified wavelengths, the matrix _{$Δ \bar{Ω}$} is then calculated using Eq. (17).

Step 3. Next, all possible arrangements for the glasses from the specified catalog are performed. For each possibility the optimum normalized power of each glass is computed using Eq. (26). The sum of the absolute power of each arrangement, given by Eq. (29) below, is used as a metric for the first weeding out. As pointed out by Rayces and Aguilar [2], high power elements have steep surfaces that result in large monochromatic aberrations, involving higher orders of aberration. This first cut eliminates potentially useless solutions. The metric used here is different from the one presented in [2]. It is more general in terms of the number of glasses used in the combination. This metric has been suggested in [11].

F_{^{1}} = \sum_{j = 1}^{k} | ϕ_{j} (λ_{0}) |

The user must set the maximum value for F₁_. The glass arrangements that have F₁ values larger than the specified value are discarded. This metric is not just used to eliminate potential useless solutions but can also be used as one of the metrics in the multi-objective approach proposed in this work. The next steps and calculations are only performed for the arrangements that comply with the F₁ limit imposed.

The vector _{$\bar{C C P}$} is than calculated by Eq. (27). The modulus of this vector, called F₂ (_{$F_{^{2}} = \bar{| C C P |}$}) can also be used in the multi-objective analysis. The smaller the value of F₂ the better the color correction the set of glasses provides as explained in Section 3.1.

Step 4. Following, a thin lens aplanatic solution for wavelength λ₀ is found for each candidate glass arrangement. To find the aplanatic solution, the system structural coefficient for spherical aberration Ξ and coma Χ are set equal to zero, with the power of each glass element calculated using Eq. (26). We ended up with the following set of equations.

Ξ = \sum_{j = 1}^{k} ξ_{j} = 0;

Χ = \sum_{j = 1}^{k} χ_{j} = 0;

\begin{matrix} [N_{1} (λ_{0}) - 1] (\frac{1}{r_{1}} - \frac{1}{r_{2}}) = (\frac{ϕ_{1} (λ_{0})}{F}) \\ ⋮ \\ [N_{k} (λ_{0}) - 1] (\frac{1}{r_{(2 k) - 1}} - \frac{1}{r_{(2 k)}}) = (\frac{ϕ_{k} (λ_{0})}{F}) \end{matrix}

To find the aplanatic solution it is necessary to solve the above set of equations for r₁ to r_2k.

For the case of a doublet, k = 2, there are four equations and four unknowns resulting in a straightforward solution. As Eq. (30) has a quadratic dependence as a function of the radius (see appendix A in [23]), two different aplanatic solutions can be obtained for each glass arrangement. The best solution is retained where the definition for a better solution is based in the metric F₃ as explained ahead.

For k≥3 there are more unknowns than equations. For solving the set of equations in an analytic and fast way, some constraint equations are added to make the number of unknowns equal to the number of equations. For example, the case where k = 3 (triplet), two options for the constraint equations are possible r₃ = r₂, or r₅ = r₄. The system can then be solved for both cases; in each case two solutions exist, which means four total possible solutions. Once more only the better solution is retained. This same idea can be expanded for k>3. The solution for the set of equations where k≥3 in not so trivial and is made with the help of a computer.

For each one of the possible retained solutions, the fifth-order spherical _{$W_{060} (λ_{0})$} and the sphero-chromatism _{$W_{040 C L} (λ_{1} \dots λ_{n})$} wave aberration coefficients are calculated according to the algorithm presented in [23]. The fifth-order spherical is calculated for the reference wavelength λ₀. The sphero-chromatism is calculated for all possible combinations of the input wavelengths, and the worse case is assigned for the set.

Step 5. The third and last metric used in the multi-objective analysis is then computed by the sum of the normalized fifth-order spherical _{${\bar{W}}_{060}$} and normalized sphero-chromatism _{${\bar{W}}_{040 C L}$} wave aberration coefficients according to Eq. (33).

F_{^{3}} = ({\bar{W}}_{040 C L} + {\bar{W}}_{060})

where [2]:

{\bar{W}}_{060} = \frac{14 \cdot W_{060} (λ_{0})}{20 \sqrt{7}}

and

{\bar{W}}_{040 C L} = \frac{14 \cdot W_{040 C L} (λ_{1} \dots λ_{n})}{6 \sqrt{5}}

The metric F₃ is also used to define which of the possible aplanatic solutions for a specific glass set is the best one, as mentioned above.

Step 6. For all the possible set of glass arrangements complying with the maximum allowed metric F₁, the best aplanatic solution is stored in a table with its respective F₁, F₂ and F₃ metric values. The data stored in the table are organized as shown in the Fig. 2 . The r’s are the radius of curvature of each surface and ϕ’s are the normalized optical power of each thin lens.

Fig. 2 Format of the table where the data for each glass arrangement best aplanatic solution is stored.

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Step 7. The solutions are then organized into different Pareto ranks using the metrics F₁, F₂ and F_3.

Step 8. At last, a post-Pareto analysis is applied in the first or in the firsts Pareto ranks, organizing the solutions in the out-put table from the best to the worse trade-off solutions.

In summary, the glass selection for the design of optical systems with reduced chromatic aberration can be seen as a multi-objective optimization problem where the goal is to minimize at the same time the objective functions F₁, F₂ and F₃, subjected to: F₁ ≤ Constant; to Eq. (30), (31) and (32), and to some additional constrains when k≥3 (e.g. r₃ = r₂, or r₅ = r_4, for the case when k = 3). The method we used here to solve the problem was an exhaustive search.

The method is also represented in a flowchart form in Fig. 3 .

Fig. 3 Flowchart of the proposed method of glass combination selection.

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4.1-Post Pareto analysis

The Pareto front, or the Pareto rank 1, specifies the global non-dominated trade-off solutions for the problem. In practice, the designer has to pick one solution from this set for designing the optical system. Despite one solution in the Pareto front not being in principle considered better than other solution in the same front, it is evident and intuitive that a discrimination among the less satisfactory trade-offs and the most promising solution can be done. This process of selecting a solution is called decision-making. Many methods for supporting this process, also known as Post-Pareto analysis, can be found in the literature [24–30].

The task of post-Pareto analysis is not so easy; especially when the number of candidate solutions is large and the number of objectives is greater than two, which is the case. Depending on the number of glasses in the catalog and the number of glasses used in the combination, hundreds of solutions are usually obtained in the Pareto front.

For this work, we used two methods of post-Pareto analysis described in sub-sections 4.1.1 and 4.1.2.

4.1.1-Minimum F₂

Organizing the solutions in the Pareto front by the most important metric is intuitive to perform the post Pareto analysis. In this paper the color correction is the most important metric, given by F₂. Rayces and Aguilar [2,23] also propose the organization of the output table of their method by the color correction index, in their case given by the secondary color. We recommend the use of this method if the number of glasses used in the set is much lower than the number of wavelengths defined and the spectral band is broad, covering different regions of the spectrum. The best glass combination is not necessarily in the first line of this table but probably among the first ones. The final choice will be made by the designer, in this case, it is important to look for a solution with a low F₃ but at the same time keeping F₂ as low as possible. Other glass parameters can also be considered in this final choice.

4.1.2-Minimum distance to the origin

Suppose a generic multi-objective problem with two objective functions O₁ and O₂_, where the goal is to minimize both functions. Suppose also that the Pareto front for this problem in the objective function space can be represented as the line plotted in Fig. 4 . This is in fact a very usual shape for a Pareto front in a min-min problem. In this Figure we highlight the “knee”, a region where the best trade-off solutions lays.

Fig. 4 Typical Pareto front for 2 objective min-min problem, showing the attributes used in the post-Pareto analysis.

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Looking at Fig. 4, it is possible to say that the bigger the length of vector g_i the less satisfactory trade-offs solution i provides. This vector g_i connects the origin of the system to a solution i on the Pareto front, having objectives values O_1i and O_2i. Due to possible different physical meanings of the objective functions, completely different numerical values ranges may be represented in each axis. This difference in range can be a problem for the use of the vector numerical length as a metric. However, we can work out this issue through the normalization of each one of the objectives. This can be done dividing O_1i by Ō₁, O_2i by Ō₂, and so on_, for each solution i. The solutions can than be organized according to the value _{$| {\bar{g}}_{i} |$}, given in its general form by Eq. (36).

| {\bar{g}}_{i} | = \sqrt{{\sum_{o b = 1}^{m} (\frac{O_{o b, i}}{{\bar{O}}_{o b}})}^{2}}

These Ō_ob values are not necessarily the highest values in the range of the solutions for each objective as we show in Fig. 4. For instance, in this work we defined this normalization factor for each variable as the value that accumulates 90% of the solutions used in the analysis. Organizing the solutions in the Pareto front in a new table using the metric given by Eq. (36), from the lowest to the highest, supports in a very nice way the decision-making. keeping the final choice for the designer that should be limited among the firsts lines in the table.

We recommend the use of this method if the number of glasses k is within n ≥ k > n/2. For the case when n is equal to k, F₂ is zero, so only functions F₁ and F₃ are used to calculate _{$| {\bar{g}}_{i} |$}. When k is lower than n, the use of only F₂ and F₃ to compute _{$| {\bar{g}}_{i} |$} is recommended. Again, the best glass combination does not necessarily lie in the very first line of this table but probably among the first ones, and the designer must make the final choice.

5. Example

In this section it is presented an example for the application of the glass selection method proposed in this work. The specification of the lens system that motivated this development, described in Section 2, is used as the example. Our intention is not to present a final design for the problem but to show how the method can be used to effectively design a multi-spectral lens system. In Table 1 it is shown the most important features specified for the optical system.

Table 1. Basic Requirements for the optical system used as example.

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The inputs for the method can be extracted from the spec in Table 1. The focal length and the F/# are taken directly from the table. For the wavelengths, the central values for each spectral band were used: 0.485, 0.55, 0.66, 0.83 and 1.6 microns. The primary wavelength λ₀ was set to 0.83 microns the due to its proximity to the central wave of the whole spectrum covered by the instrument.

The newest available Schott glass catalog was selected [31] to run the method. However, some specific glasses from this catalog were discarded: Lithotec-CAF2, N-PK51, N-PK52A, N-FK51A, P-PK53, N-PSK53A and N-PSK53. Despite these glasses being very good options for color correction, they were rejected due to their undesirable thermal behavior. Optical systems designed with these glasses are potentially sensitive to temperature changes. Normally, for small changes of temperature, the effect can be compensated with refocusing, however, the application of the instrument object of this example, cannot afford either a manual or an automatic refocus mechanism.

At first we ran the method for arrangements of two glasses. The limit F₁ defined for this case was 9. The post-Pareto analysis was applied only for the solutions in the Pareto ranking 1, using the method presented in Section 4.1.1.

In Table 2 we can see the first 10 rows of the output table, sorted from the smallest to the biggest F₂.

Table 2. Output table from the glass selection method for 2 glasses sorted by F₂.

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We believe that the fourth line of Table 2 brings the best trade-off option for the combination of 2 glasses for the problem. Solutions above the fourth line have F₂ values slightly smaller, however, the F₃ values are significantly higher. Bellow the forth line the F₂ values increase very fast.

Before we go for the design of the optical system with the selected pair of glass, we can perform a roughly check to see if it is promising in terms of the color correction. It is known that the tolerable depth of focus of a system can be given by [19]:

ε = \pm 2 λ {(f #)}^{2}

Calculating Eq. (37), using the f# provided in the Table 1 and the selected primary wavelength λ₀ (0.83μm), results in ε = ± 0.0415mm. We can compare this value to the result from the multiplication between F₂ and the focal length F, which for the selected pair gives 0.184mm. This number is much higher than the calculated ε, telling us that the design with the selected pair is not promising. Even with the lowest F₂ value shown in the first line of Table 2, we cannot even get close to the calculated ε. The conclusion is that more glasses to the set are necessary to design the desirable system with the glass catalog used.

In this case, we ran the method again with the same parameters but now for three glasses in the set. With more glasses, the F₁ limit was changed to 11. The most suitable post-Pareto method in this situation is the one presented in Section 4.1.2, where the metric _{$| {\bar{g}}_{i} |$} is calculated using only F₂ and F₃.

In Table 3 we can see the first 15 rows of the output table resulted from the application of the method using 3 glasses in the set. The solutions are sorted from the smallest to the biggest _{$| {\bar{g}}_{i} |$}.

Table 3. Output table from the glass selection method for 3 glasses sorted by_{$| {\bar{g}}_{i} |$}.

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For the case of three glasses, we selected two solutions among the first lines that we believe to be good trade-offs. The chosen solutions are located in the fifth and thirteenth lines of Table 3. The first one has a smaller F₃ than the second one and also a better power distribution among the lenses. On the other hand the second has a smaller F₂. Calculating the multiplication between F₂ and the focal length for both solutions we get 0.047mm and 0.014mm respectively. These numbers reveals that these glass combinations are promising in terms of color correction.

Figure 5 shows the chromatic focal shift for the two aplanatic thin triplet obtained from the glass combination chosen from Table 3. The one in the left side of the Fig. 5 corresponds to the glasses on the fifth line of Table 3 (N-BAF52, N-KZFS11 and N-BAK2) while the one in the right side corresponds to the ones on the thirteenth line of Table 3 (N-KZFS8, P-SF68 and N-SK2). This graphs reveals that despite not crossing the axis five times in the center of all spectral bands, the shift is not greater than 33 microns for the combination N-BAF52, N-KZFS11 and N-BAK2 and less than 13 microns for the combination N-KZFS8, P-SF68 and N-SK2 for the central wavelength of each spectral band. This gives us confidence that the design of the objective can be done. The residual chromatic focal shift can be compensated with a slightly change on the position of each spectral band image plane, as each band will focus in a different detector. In this case we can go for the design.

Fig. 5 Chromatic focal Shift for the aplanatic triplets designed with glass combination (a) N-BAF52, N-KZFS11 and N-BAK2, and combination (b) N-KZFS8, P-SF68 and N-SK2.

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The output glass combination chosen after the application of the method proposed can then be used to design an optical system either applying classical or evolutionary methods. In this last one the advantage is the significantly reduction of the design space, decreasing the number of glasses from thousands to just a few.

The lens design lay out for the glass combination N-BAF52, N-KZFS11 and N-BAK2 can be seen in the top left side of Fig. 6 , where each lens glass is identified. The system complies with all the basic requirements presented in Table 3. The image quality is also shown in Fig. 6 through the MTF curves. The Quality is fair for the blue band and great for the other bands.

Fig. 6 Layout (a) and MTFs (b)(c)(d)(e)(f) for each spectral band and field position for the design made with glass combination N-BAF52, N-KZFS11 and N-BAK2.

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In Fig. 7 the layout for the lens system designed with the glass combination N-KZFS8, P-SF68 and N-SK2 is shown in the top left side. The glass of each one of the lenses is identified in the layout. Notice that the glass P-SF68 is present only in one lens. This reflects the big difference in the power distribution between the 2 positive lenses as shown in Table 3.

Fig. 7 Layout (a) and MTFs (b)(c)(d)(e)(f) for each spectral band and field position for the design made with glass combination N-KZFS8 P-SF68 and N-SK2.

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The MTF curves for each one of the spectral bands for this system is also presented in Fig. 7. Again the image quality is fair for the blue band and excellent for the other bands for the whole field of view.

For both systems presented, the MTF curves reported were obtained with each spectral band focusing in its best focus. The prescription data for the lens shown in Fig. 6 and Fig. 7 are presented in Table 4 and Table 5 in the end of this section. The systems were designed using only spherical lenses.

Table 4. Prescription data for the system shown in Fig. 6.

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Table 5. Prescription data for the system shown in Fig. 7.

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Despite being very good designs, the systems presented in this example might not represent final designs for the system that motivated the development of the glass selection method proposed herein. Probably more elements in the system will be needed in order to comply with all the detailed optical requirements necessary for the system, as well as to accommodate the beam splitters necessary for the spectral bands separation in the different detectors. Although, care was taken to design systems that are very representative in order to show the feasibility of the project. For example we avoided the use of some glasses with potential thermal problems, and also cemented lens that would facilitate the design, controlling easier the lateral color, but would not be desired for the final system due to some thermo mechanical constraints.

6. Conclusion

In the research of available methods and techniques of glass selection for color correction in lens design, we realized that all presented approaches in the literature had some drawbacks and/or missing points. During the literature survey two of the reviewed methods called our attention: the Mercado and Robb method [7] and the Rayces and Aguilar method [2]. The first one presents a technique in a very general form in terms of the number of wavelengths where the achromatization is desired, spectral region, and number of glass material used in the set. However, it has some practical implementation issues that limit its use to some specific cases. The second is limited to the combination of only two glasses and three wavelengths. Nevertheless, it proposes the use of some metrics very important in the identification of promising glass combinations that can potentially provide good final designs.

Unifying these two mention methods, providing some original contributions to Mercado and Robb technique that repair its practical issues, and using a multi-objective approach, a new method of glass selection for color correction was developed. This new method offers significant advantages in the task of glass selection, leading to optimum choice of optical glasses for specific problems, as pointed out and demonstrated here.

Along this paper we went from the background theory, passing through a detailed description of the new proposed method and finally wrapped it up with a practical example. The design examples demonstrated the power of the proposed method in fiddling compatible glasses that are able to conduct to excellent final designs. The results of this paper offer significant improvements to the problem of glass selection in lens design when color correction is an important matter, converting this task into a systematic and objective work. The efficiency of this new method will be investigated in future papers for other lens design problems, involving different spectral bands and types of lens systems.

We intend to incorporate the presented method of glass selection together with evolutionary optimization methods in lens design that we have been working on. The glass selection method has the ability to identify the most promising glasses to be used in a certain design, reducing the glass options from hundreds to just a few. As a consequence the design space is reduced and simplified, what is a significant advantage for global search heuristic methods.

In a near future, free stand-alone version of the software with the method presented here will be provided. The software will be found in the site: http://www.optics.arizona.edu/glassselectiontool. Meanwhile, we can provide free of charge tables of glass combinations upon request for specific input data provided. The request can be done by e-mail: braulio@dea.inpe.br.

Acknowledgments

B. F. C. Albuquerque gratefully acknowledges the Brazilian Research Council (CNPq) for supporting these studies and the Kidger Optics Associates for also supporting these studies through the 2010 Michael Kidger Memorial Scholarship in Optical Design.

References and links

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31. SCHOTT N. America, Inc., “Optical glass catalogue- ZEMAX format, status as of 13th September 2011, http://www.us.schott.com/advanced_optics/english/tools_downloads/download/index.html?PHPSESSID=utt2cbk96nlk3gf7gjpb7ggt54#Optical%20Glass

ITEM	REQUIREMENT
Effective Focal Length	250mm
F/ Number (f#)	5
Field of view	± 9 degrees
Spectral Bands	0.450-0.520; 0.520-0.590; 0.630-0.690;  0.770-0.890 e 1.5-1.7μm
Maximal Distortion	3%
Field relative illumination	Constant ± 3%
Back focal length (BFL)	Large enough to fit the spectral bands beamsplitter (more than 50mm).
Design Resolution (MTF)	Close to diffraction limit for all bands in sagittal and tangential directions.

N°	Glass 1	Glass 2	r₁	r₂	r₃	r₄	φ₁	φ₂	F₁	F₂	F₃
771	N-BALF4	N-KZFS11	170.91	−37.58	−38.36	−351.12	4.63	−3.63	8.27	5.46E-04	16.28
606	N-BAK1	N-KZFS11	163.36	−47.77	−48.80	−403.20	3.82	−2.82	6.64	5.75E-04	8.69
1383	N-KZFS11	N-BAK1	111.63	37.08	36.53	3124.93	−2.82	3.82	6.64	5.75E-04	8.59
1417	N-KZFS11	N-SK2	129.72	38.16	37.98	3141.56	-2.90	3.90	6.79	7.36E-04	6.94
4394	N-SSK5	KZFS12	168.54	−58.44	−58.88	−1078.81	3.74	−2.74	6.47	2.50E-03	6.12
216	KZFS12	N-SSK5	134.49	42.57	42.37	1796.69	−2.74	3.74	6.47	2.50E-03	5.91
2092	N-LAK12	KZFS12	168.51	−74.49	−74.20	−2741.16	3.23	−2.23	5.47	3.72E-03	4.35
191	KZFS12	N-LAK12	145.49	50.04	50.19	1765.35	−2.23	3.23	5.47	3.72E-03	4.29
212	KZFS12	N-SK2	118.05	48.24	47.64	2732.27	−2.09	3.09	5.18	3.80E-03	4.32
596	N-BAK1	KZFS12	160.66	−65.53	−67.81	−355.23	3.03	−2.03	5.07	4.20E-03	4.92

N°	Glass 1	Glass 2	Glass 3	r₁	•••	r₆	φ₁	φ₂	φ₂	F₁	F₂	F₃
5317	N-BAF52	N-KZFS11	N-SK4	145.26	•••	−1056.62	3.12	−4.68	2.56	10.37	2.15E-04	2.98
35786	N-SK4	N-KZFS11	N-BAF52	135.68	•••	−2128.74	2.56	−4.68	3.12	10.37	2.15E-04	2.99
11387	N-KZFS11	N-BAF52	N-SK4	−42.02	•••	−55.13	−4.68	3.12	2.56	10.37	2.15E-04	3.03
25496	N-LAK8	N-KZFS11	N-BAK2	−20.89	•••	−23.05	−3.21	−1.58	5.79	10.59	4.61E-04	1.49
5310	N-BAF52	N-KZFS11	N-BAK2	110.35	•••	-1266.01	3.34	-4.71	2.38	10.43	1.88E-04	3.14
26091	N-LASF31A	N-KZFS2	N-BAK1	−16.58	•••	−17.83	−2.09	−2.03	5.13	9.26	5.11E-04	0.67
13718	N-KZFS11	N-SSK8	N-SK4	−45.56	•••	−62.64	−3.71	2.41	2.29	8.41	4.28E-04	2.04
13672	N-KZFS11	N-SSK8	N-BAK2	−47.45	•••	−65.80	−3.67	2.56	2.11	8.35	4.17E-04	2.14
9960	N-FK5	N-KZFS11	N-BAF51	165.17	•••	−264.64	2.02	−4.37	3.35	9.74	3.18E-04	2.84
11541	N-KZFS11	N-BAK2	N-SSK8	−43.54	•••	−58.25	−3.67	2.11	2.56	8.35	4.17E-04	2.23
6824	N-BAK2	N-KZFS11	N-SSK8	142.16	•••	−444.22	2.11	−3.67	2.56	8.35	4.17E-04	2.28
37966	N-SSK8	N-KZFS11	N-BAK2	115.23	•••	−1534.96	2.56	−3.67	2.11	8.35	4.17E-04	2.31
20485	N-KZFS8	P-SF68	N-SK2	98.06	•••	812.98	-4.73	1.17	4.56	10.46	5.66E-05	3.56
2283	KZFS12	N-SF4	N-SK4	112.05	•••	1464.73	−4.48	1.54	3.94	9.96	3.67E-04	2.68
11237	N-KZFS11	LLF1	N-SK14	−42.00	•••	−54.57	−4.20	2.49	2.70	9.39	3.75E-04	2.63

System 1. EFL = 250mm; F/# = 5.
Surf	Radius (mm)	Thickness (mm)	Glass
OBJ	Infinity	Infinity
1	114.506	7.737	N-BAF52
2	1551.850	2.887
3	−787.580	8.220	N-KZFS11
4	58.764	2.534
5	60.880	8.455	N-BAK2
6	173.824	59.804
STO	Infinity	20.929
8	125.540	10.496	N-BAF52
9	−80.372	2.299
10	−75.012	7.000	N-KZFS11
11	84.061	2.084
12	84.729	12.000	N-BAK2
13	−170.627	126.151
14	−69.911	7.833	N-KZFS11
15	−207.232	50.887

System 2. EFL = 250mm; F/# = 5.
Surf	Radius (mm)	Thickness (mm)	Glass
OBJ	Infinity	Infinity
1	120.751	6.425	N-SK2
2	−1753.068	4.809
3	−112.487	6.002	N-KZFS8
4	179.109	1.011
5	115.631	8.594	N-SK2
6	−119.451	0.130
STO	Infinity	11.169
8	12071.420	6.000	P-SF68
9	−141.327	2.586
10	−107.188	8.000	N-KZFS8
11	62.025	3.207
12	68.587	12.000	N-SK2
13	159.482	40.000
14	163.086	12.000	N-SK2
15	−257.749	98.105
16	−82.596	12.000	N-KZFS8
17	−253.127	50.000

Method of glass selection for color correction in optical system design

Abstract

1. Introduction

2. Motivation