## Method of glass selection for color correction in optical system design |

Optics Express, Vol. 20, Issue 13, pp. 13592-13611 (2012)

http://dx.doi.org/10.1364/OE.20.013592

Acrobat PDF (1752 KB)

### Abstract

A method of glass selection for the design of optical systems with reduced chromatic aberration is presented. This method is based on the unification of two previously published methods adding new contributions and using a multi-objective approach. This new method makes it possible to select sets of compatible glasses suitable for the design of super-apochromatic optical systems. As an example, we present the selection of compatible glasses and the effective designs for all-refractive optical systems corrected in five spectral bands, with central wavelengths going from 485 nm to 1600 nm.

© 2012 OSA

## 1. Introduction

2. J. L. Rayces and M. Rosete-Aguilar, “Selection of glasses for achromatic doublets with reduced secondary spectrum. I. Tolerance conditions for secondary spectrum, spherochromatism, and fifth-order spherical aberration,” Appl. Opt. **40**(31), 5663–5676 (2001). [CrossRef] [PubMed]

3. R. D. Sigler, “Glass selection for airspaced apochromats using the Buchdahl dispersion equation,” Appl. Opt. **25**(23), 4311–4320 (1986). [CrossRef] [PubMed]

2. J. L. Rayces and M. Rosete-Aguilar, “Selection of glasses for achromatic doublets with reduced secondary spectrum. I. Tolerance conditions for secondary spectrum, spherochromatism, and fifth-order spherical aberration,” Appl. Opt. **40**(31), 5663–5676 (2001). [CrossRef] [PubMed]

13. R. E. Stephens, “Four-color achromats and superchromats,” J. Opt. Soc. Am. **50**(10), 1016–1019 (1960). [CrossRef]

14. W. S. Sun, C. H. Chu, and C. L. Tien, “Well-chosen method for an optimal design of doublet lens design,” Opt. Express **17**(3), 1414–1428 (2009). [CrossRef] [PubMed]

15. I. Ono, Y. Tatsuzawa, S. Kobayashi, and K. Yoshida, “Designing lens systems taking account of glass selection by real-coded genetic algorithms,” in *Proceedings of IEEE International Conference on Systems, Man and Cybernetics* (Institute of Electrical and Electronics Engineers, New York, 1999), 7803–5731.

16. Y. C. Fang, C. M. Tsai, J. Macdonald, and Y. C. Pai, “Eliminating chromatic aberration in Gauss-type lens design using a novel genetic algorithm,” Appl. Opt. **46**(13), 2401–2410 (2007). [CrossRef] [PubMed]

17. L. Li, Q. H. Wang, X. Q. Xu, and D. H. Li, “Two-step method for lens system design,” Opt. Express **18**(12), 13285–13300 (2010). [CrossRef] [PubMed]

*et al*[18

18. R. E. Fischer, A. J. Grant, U. Fotheringham, P. Hartmann, and S. Reichel, “Removing the mystique of glass selection,” Proc. SPIE **5524**, 134–146 (2004). [CrossRef]

2. J. L. Rayces and M. Rosete-Aguilar, “Selection of glasses for achromatic doublets with reduced secondary spectrum. I. Tolerance conditions for secondary spectrum, spherochromatism, and fifth-order spherical aberration,” Appl. Opt. **40**(31), 5663–5676 (2001). [CrossRef] [PubMed]

## 2. Motivation

**40**(31), 5663–5676 (2001). [CrossRef] [PubMed]

9. N. V. D. W. Lessing, “Selection of optical glasses in superachromats,” Appl. Opt. **9**(7), 1665–1668 (1970). [CrossRef] [PubMed]

11. M. Herzberger and N. R. McClure, “The design of superachromatic lenses,” Appl. Opt. **2**(6), 553–560 (1963). [CrossRef]

16. Y. C. Fang, C. M. Tsai, J. Macdonald, and Y. C. Pai, “Eliminating chromatic aberration in Gauss-type lens design using a novel genetic algorithm,” Appl. Opt. **46**(13), 2401–2410 (2007). [CrossRef] [PubMed]

## 3. Background of the proposed method

**40**(31), 5663–5676 (2001). [CrossRef] [PubMed]

13. R. E. Stephens, “Four-color achromats and superchromats,” J. Opt. Soc. Am. **50**(10), 1016–1019 (1960). [CrossRef]

4. C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. **40**(2), 131–140 (2008). [CrossRef]

6. P. Hariharan, “Apochromatic lens combinations, a novel design approach,” Opt. Laser Technol. **29**(4), 217–219 (1997). [CrossRef]

9. N. V. D. W. Lessing, “Selection of optical glasses in superachromats,” Appl. Opt. **9**(7), 1665–1668 (1970). [CrossRef] [PubMed]

11. M. Herzberger and N. R. McClure, “The design of superachromatic lenses,” Appl. Opt. **2**(6), 553–560 (1963). [CrossRef]

13. R. E. Stephens, “Four-color achromats and superchromats,” J. Opt. Soc. Am. **50**(10), 1016–1019 (1960). [CrossRef]

### 3.1-The Mercado and Robb method with some new contributions

20. 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]

*N*represents the refraction index for wavelength

*λ*.

*N*

_{0}is the refraction index in a reference wavelength

*λ*, and

_{0}*ω*is a function of the wavelength

*λ*that is called chromatic coordinate:where

*δλ*=

*λ-λ*, and

_{0}*α*is a universal constant taken as 2.5 [7]. The dispersion coefficients

*ν*, 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

_{n}20. 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]

*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*of each glass were the number of unknowns is equal to the number of equations.

_{n}*N*

_{0}to the left side of Eq. (1) and by dividing both sides by the constant

*N*–1, we obtain:where:

_{0}_{D(λ)=δN(λ)/(N0−1)};

_{δN(λ)=N(λ)−N0}and

_{ηi=νi/(N0−1)}. The term

*D(λ)*is called dispersive power.This equation is very important since the method presented in [7] is mainly based on it.

*ϕ*of a lens is defined as the inverse of the it’s focal length

*f*:

*C*and

_{1}*C*are the lens curvature.

_{2}*C*) must be a constant, conveniently called

_{1}– C_{2}*K*. Thus we can write:

_{0}can be expressed by:

_{D(λ)=δN(λ)/(N0−1)}, and

_{δN(λ)=N(λ)−N0}, together with Eq. (7), we can write for the optical power:

*k*thin lenses in contact, the resulting optical power for the reference wavelength

*λ*is computed by:

_{0}*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:

*λ*), Φ(

_{1}*λ*), Φ(

_{2}*λ*), ..., Φ(

_{3}*λ*) can be transcribed in the following form:

_{n}*j*over the wavelength range

*λ*<

_{1}*λ*<

*λ*, can be written in simplified form as:

_{2}*ω*as:

_{ΔΩ¯}is a square matrix of order

*n-*1 x

*n-*1:

_{η¯}is a matrix of order

*n-*1 x

*k*:

_{Φ¯}is a matrix of order

*k*x1:and

_{0¯}is a matrix of the

*n-*1 x 1 order:

_{ΔΩ¯}is a square and doubtless nonsingular. As a consequence its inverse

_{(ΔΩ¯−1)}exists. Multiplying both sides of Eq. (16) by

_{ΔΩ¯−1}results in a condition to obtain a solution free from chromatic aberration for all the wavelengths defined:

_{Φ¯≠0¯}) if and only if the matrix

_{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

_{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

_{η¯}rank will always be equal to

*k*. This result makes the rank of matrix

_{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.

*λ*. As a consequence Eq. (9) becomes:

_{0}_{S¯}is a row vector of order 1x

*k*, with all elements equal to one.

_{e^}is a column vector of order

*n*x1 with the first element equal to one and the others zero as shown below:

_{Φ¯^}in Eq. (16), to obtain the minimum chromatic change of power

_{CCP¯}, as expressed in Eq. (27).

_{CCP¯}.

_{CCP¯}by the desired effective focal length

*F*for the optical system.As the chromatic focal shift is proportional to

_{CCP¯}it clearly gives physical meaning to our metric.

### 3.2-The Rayces-Aguilar method

### 3.3-The multi-objective approach

*F*and

_{1}*F*.

_{2}*F*and

_{1}*F*the better the solution is.

_{2}22. N. Srinivas and K. Deb, “Multi-objective function optimization using non-dominated sorting genetic algorithm,” Evol. Comput. **2**(3), 221–248 (1994). [CrossRef]

## 4. The synthesis method of glass selection

**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

*λ*. A glass catalog and the number of glasses used in the combination (i.e. 2, 3, 4, etc) must also be specified.

_{0}**Step 2.**At the outset, the first

*n-*1 dispersion coefficients

*η*, are calculated for each glass in the catalog. For that, the

_{i}*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*dispersion coefficients. With the specified wavelengths, the matrix

_{i}_{ΔΩ¯}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

**40**(31), 5663–5676 (2001). [CrossRef] [PubMed]

**40**(31), 5663–5676 (2001). [CrossRef] [PubMed]

11. M. Herzberger and N. R. McClure, “The design of superachromatic lenses,” Appl. Opt. **2**(6), 553–560 (1963). [CrossRef]

*F*

_{1}_{.}The glass arrangements that have

*F*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

_{1}*F*limit imposed.

_{1}_{CCP¯}is than calculated by Eq. (27). The modulus of this vector, called

*F*(

_{2}_{F2=|CCP|¯}) can also be used in the multi-objective analysis. The smaller the value of

*F*the better the color correction the set of glasses provides as explained in Section 3.1.

_{2}**Step 4.**Following, a thin lens aplanatic solution for wavelength

*λ*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.

_{0}*r*to r

_{1}*.*

_{2k}*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

23. J. Rayces and M. R. Aguilar, “Selection of glasses for achromatic doublets with reduced secondary color,” Proc. SPIE **4093**, 36–46 (2000). [CrossRef]

*F*as explained ahead.

_{3}*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

*r*, or

_{3}= r_{2}*r*. 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

_{5}= r_{4}*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.

_{W060(λ0)}and the sphero-chromatism

_{W040CL(λ1⋯λn)}wave aberration coefficients are calculated according to the algorithm presented in [23

23. J. Rayces and M. R. Aguilar, “Selection of glasses for achromatic doublets with reduced secondary color,” Proc. SPIE **4093**, 36–46 (2000). [CrossRef]

*λ*. The sphero-chromatism is calculated for all possible combinations of the input wavelengths, and the worse case is assigned for the set.

_{0}**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

_{W¯060}and normalized sphero-chromatism

_{W¯040CL}wave aberration coefficients according to Eq. (33).where [2

**40**(31), 5663–5676 (2001). [CrossRef] [PubMed]

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

_{3}**Step 6.**For all the possible set of glass arrangements complying with the maximum allowed metric

*F*, the best aplanatic solution is stored in a table with its respective

_{1}*F*,

_{1}*F*and

_{2}*F*metric values. The data stored in the table are organized as shown in the Fig. 2 . The

_{3}*r’s*are the radius of curvature of each surface and

*ϕ’s*are the normalized optical power of each thin lens.

**Step 7**. The solutions are then organized into different Pareto ranks using the metrics

*F*,

_{1}*F*and

_{2}*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.

*F*,

_{1}*F*and

_{2}*F*, subjected to:

_{3}*F*≤

_{1}*Constant*; to Eq. (30), (31) and (32), and to some additional constrains when

*k*≥3 (e.g.

*r*, or

_{3}= r_{2}*r*for the case when

_{5}= r_{4,}*k = 3*). The method we used here to solve the problem was an exhaustive search.

### 4.1-Post Pareto analysis

24. N. Lopez, O. Aguirre, J. F. Espiritu, and H. A. Taboada, “Using game theory as a post-Pareto analysis for renewable energy integration problems considering multiple objectives,” in *Proceedings of the 41st International Conference on Computers & Industrial Engineering*, 678–683 Los Angeles, (2011).

#### 4.1.1-Minimum F_{2}

*F*. Rayces and Aguilar [2

_{2}**40**(31), 5663–5676 (2001). [CrossRef] [PubMed]

23. J. Rayces and M. R. Aguilar, “Selection of glasses for achromatic doublets with reduced secondary color,” Proc. SPIE **4093**, 36–46 (2000). [CrossRef]

*F*but at the same time keeping

_{3}*F*as low as possible. Other glass parameters can also be considered in this final choice.

_{2}#### 4.1.2-Minimum distance to the origin

*O*and

_{1}*O*

_{2}_{,}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.

*g*the less satisfactory trade-offs solution

_{i}*i*provides. This vector

*g*connects the origin of the system to a solution

_{i}*i*on the Pareto front, having objectives values

*O*and

_{1i}*O*. 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

_{2i}*O*by

_{1i}*Ō*,

_{1}*O*by

_{2i}*Ō*, and so on

_{2}_{,}for each solution

*i*. The solutions can than be organized according to the value

_{|g¯i|}, given in its general form by Eq. (36).

*Ō*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.

_{ob}*k*is within

*n*≥

*k*>

*n*/2. For the case when

*n*is equal to

*k*,

*F*is zero, so only functions

_{2}*F*and

_{1}*F*are used to calculate

_{3}_{|g¯i|}. When

*k*is lower than

*n*, the use of only

*F*and

_{2}*F*to compute

_{3}_{|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

*λ*was set to 0.83 microns the due to its proximity to the central wave of the whole spectrum covered by the instrument.

_{0}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

*F*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.

_{1}*F*.

_{2}*F*values slightly smaller, however, the

_{2}*F*values are significantly higher. Bellow the forth line the

_{3}*F*values increase very fast.

_{2}*λ*(0.83μm), results in

_{0}*ε*= ± 0.0415mm. We can compare this value to the result from the multiplication between

*F*and the focal length

_{2}*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

*F*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.

_{2}*F*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

_{1}_{|g¯i|}is calculated using only

*F*and

_{2}*F*.

_{3}_{|g¯i|}.

*F*than the second one and also a better power distribution among the lenses. On the other hand the second has a smaller

_{3}*F*. Calculating the multiplication between

_{2}*F*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.

_{2}## 6. Conclusion

**40**(31), 5663–5676 (2001). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | P. Mouroulis, “Broadband achromatic telecentric lens,” Nasa Tech Briefs, |

2. | J. L. Rayces and M. Rosete-Aguilar, “Selection of glasses for achromatic doublets with reduced secondary spectrum. I. Tolerance conditions for secondary spectrum, spherochromatism, and fifth-order spherical aberration,” Appl. Opt. |

3. | R. D. Sigler, “Glass selection for airspaced apochromats using the Buchdahl dispersion equation,” Appl. Opt. |

4. | C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. |

5. | P. Hariharan, “Superachromatic lens combination,” Opt. Laser Technol. |

6. | P. Hariharan, “Apochromatic lens combinations, a novel design approach,” Opt. Laser Technol. |

7. | R. I. Mercado and P. N. Robb, “Color corrected optical systems and method of selecting optical materials therefor,” U.S Patent, |

8. | P. N. Robb, “Selection of optical glasses. 1: two materials,” Appl. Opt. |

9. | N. V. D. W. Lessing, “Selection of optical glasses in superachromats,” Appl. Opt. |

10. | T. R. Sloan, “Analysis and correction of secondary color in optical systems,” Appl. Opt. |

11. | M. Herzberger and N. R. McClure, “The design of superachromatic lenses,” Appl. Opt. |

12. | R. R. Willey Jr., “Machine-aided selection of optical glasses for two-elements, three-color achromats,” Appl. Opt. |

13. | R. E. Stephens, “Four-color achromats and superchromats,” J. Opt. Soc. Am. |

14. | W. S. Sun, C. H. Chu, and C. L. Tien, “Well-chosen method for an optimal design of doublet lens design,” Opt. Express |

15. | I. Ono, Y. Tatsuzawa, S. Kobayashi, and K. Yoshida, “Designing lens systems taking account of glass selection by real-coded genetic algorithms,” in |

16. | Y. C. Fang, C. M. Tsai, J. Macdonald, and Y. C. Pai, “Eliminating chromatic aberration in Gauss-type lens design using a novel genetic algorithm,” Appl. Opt. |

17. | L. Li, Q. H. Wang, X. Q. Xu, and D. H. Li, “Two-step method for lens system design,” Opt. Express |

18. | R. E. Fischer, A. J. Grant, U. Fotheringham, P. Hartmann, and S. Reichel, “Removing the mystique of glass selection,” Proc. SPIE |

19. | W. J. Smith, |

20. | P. N. Robb and R. I. Mercado, “Calculation of refractive indices using Buchdahl’s chromatic coordinate,” Appl. Opt. |

21. | J. Branke, K. Deb, K. Miettinen, and R. Slowinski, |

22. | N. Srinivas and K. Deb, “Multi-objective function optimization using non-dominated sorting genetic algorithm,” Evol. Comput. |

23. | J. Rayces and M. R. Aguilar, “Selection of glasses for achromatic doublets with reduced secondary color,” Proc. SPIE |

24. | N. Lopez, O. Aguirre, J. F. Espiritu, and H. A. Taboada, “Using game theory as a post-Pareto analysis for renewable energy integration problems considering multiple objectives,” in |

25. | O. Aguirre, H. Taboada, D. Coit, and N. Wattanapongsakorn, “Multiple objective system reliability post-Pareto optimality using self organizing trees,” in |

26. | E. Zio and R. Bazzo, “Clustering procedure for reducing the number of representative solutions in the Pareto front of multiobjective optimization problems,” Eur. J. Oper. Res. |

27. | X. Blasco, J. M. Herrero, J. Sanchis, and M. Martínez, “A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization,” Inf. Sci. |

28. | J. C. Ferreira, C. M. Fonseca, and A. Gaspar-Cunha, “Methodology to select solutions from the Pareto-optimal set: A comparative study,” in |

29. | V. Venkat, S. H. Jacobson, and J. A. Stori, “A Post-optimality analysis algorithm for multi-objective optimization,” Comput. Optim. Appl. |

30. | C. A. Coello Coello, “Handling preferences in evolutionary multiobjective optimization: a survey,” in |

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 |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(080.2720) Geometric optics : Mathematical methods (general)

(080.3620) Geometric optics : Lens system design

(220.0220) Optical design and fabrication : Optical design and fabrication

(220.1000) Optical design and fabrication : Aberration compensation

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: February 8, 2012

Revised Manuscript: March 27, 2012

Manuscript Accepted: March 27, 2012

Published: June 4, 2012

**Citation**

Bráulio Fonseca Carneiro de Albuquerque, Jose Sasian, Fabiano Luis de Sousa, and Amauri Silva Montes, "Method of glass selection for color correction in optical system design," Opt. Express **20**, 13592-13611 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-13592

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### References

- P. Mouroulis, “Broadband achromatic telecentric lens,” Nasa Tech Briefs, NPO-44059, (2007).
- J. L. Rayces and M. Rosete-Aguilar, “Selection of glasses for achromatic doublets with reduced secondary spectrum. I. Tolerance conditions for secondary spectrum, spherochromatism, and fifth-order spherical aberration,” Appl. Opt. 40(31), 5663–5676 (2001). [CrossRef] [PubMed]
- R. D. Sigler, “Glass selection for airspaced apochromats using the Buchdahl dispersion equation,” Appl. Opt. 25(23), 4311–4320 (1986). [CrossRef] [PubMed]
- C. Gruescu, I. Nicoara, D. Popov, R. Bodea, and H. Hora, “Optical glass compatibility for the design of apochromatic systems,” Sci. Sin. 40(2), 131–140 (2008). [CrossRef]
- P. Hariharan, “Superachromatic lens combination,” Opt. Laser Technol. 31(2), 115–118 (1999). [CrossRef]
- P. Hariharan, “Apochromatic lens combinations, a novel design approach,” Opt. Laser Technol. 29(4), 217–219 (1997). [CrossRef]
- R. I. Mercado and P. N. Robb, “Color corrected optical systems and method of selecting optical materials therefor,” U.S Patent, 5,210,646, (1993).
- P. N. Robb, “Selection of optical glasses. 1: two materials,” Appl. Opt. 24(12), 1864–1877 (1985). [CrossRef] [PubMed]
- N. V. D. W. Lessing, “Selection of optical glasses in superachromats,” Appl. Opt. 9(7), 1665–1668 (1970). [CrossRef] [PubMed]
- T. R. Sloan, “Analysis and correction of secondary color in optical systems,” Appl. Opt. 9(4), 853–858 (1970). [CrossRef] [PubMed]
- M. Herzberger and N. R. McClure, “The design of superachromatic lenses,” Appl. Opt. 2(6), 553–560 (1963). [CrossRef]
- R. R. Willey., “Machine-aided selection of optical glasses for two-elements, three-color achromats,” Appl. Opt. 1(3), 368–369 (1962). [CrossRef]
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