Abstract

An imaging design approach for optical systems consisting of two aspheres which is free of astigmatism is presented in this paper. A set of implicit differential equations is derived from generalized ray tracing. The solution of the derived equations provides the profiles of the two aspheres as well as the object to image mapping. The obtained design can be used as a good starting point for optimization. Particular examples are given.

© 2015 Optical Society of America

1. Introduction

Aspheric surfaces can in general provide more compact imaging designs with better performance than their spherical counterparts. We call aspheric surface or asphere to a rotational symmetric surface which is not a portion of a sphere.

Schmidt corrector plate is an early example of a single aspheric design to correct spherical aberration in a telescope [1], and Schwarzschild pioneered the design of two aspheres for an aplanatic system [2]. Later, Wassermann and Wolf [3] generalized Schwarzschild approach presenting a method for the design of two aspheres as solution of two simultaneous first-order ordinary differential equations. Standard integrating methods have been applied to obtain the aspheric profiles numerically. More recently Willstrop and Lynden-Bell gave general analytic solutions of the two-mirror aplanatic design problem [4, 5]. The 2D-SMS (Two-dimensional Simultaneous Multiple Surfaces) method, which can be seen as a system of functional differential equations, is another design procedure that has been used to design directly two or more aspheres for imaging [6]. It has been used, for instance, to design a telephoto for the SWIR band [7]. In this design the 6 SMS aspheres fitted with Forbes polynomials [8, 9] were used as the starting point of an optimization procedure. As yet there are 2D-SMS designs perfectly imaging up to 8 fields. This method can also be used to image several fields using less aspheres, only by controlling the size of the pupil aperture [10].

Novel direct methods for designing an anastigmatic (free of astigmatism) single-asphere and an anastigmatic single-freeform-surface optical systems have been recently presented [11, 12]. These methods are based on finding a differential equation on the optical surface deduced by equating the principal curvatures of the output wavefronts. In this design procedure, the mapping between object and image points cannot be prescribed. This mapping and the shape of the image surface are obtained as a result of the design process. Here, we have extended this strategy to the design of a double asphere anastigmatic optical system. The second surface provides additional freedom and, unlike the preceding single surface method, now both the object and the image surface shapes can be prescribed. The derivation of the differential equations is done in section 3. In section 4, an anastigmatic design is presented and compared with an aplanatic design as a starting point for multi-parameter optimization.

2. Statement of the problem

The design method introduced here adopts the definition of “supporting” and “supported” wavefronts from [11]. The input supporting wavefront is normal to a set of rays in the object space, which once refracted or reflected by the two unknown aspheres, define the output supporting wavefront in the image space. There is a supported wavefront per each ray of the supporting wavefront. This ray is also normal to its corresponding supported wavefront. Then, the supported wavefronts are tangent to the supporting wavefront, but with different principal curvatures at the points of tangency. In our approach, we will consider a second order approximation of the supported wavefronts in the neighborhood of the point tangent to the supporting wavefront, therefore, each supported wavefront surface will be fully characterized by the tangent point, the ray direction and the value of the principal curvatures relative to those of the supporting wavefront, all these values taken at the tangent point. The design problem for double surface system is to find the optical surfaces (refractive or reflective) such that the second order approximations of the output supported wavefronts when they get the image are spheres centered at the image points, so the design is free from astigmatism. The image surface, where the centers of curvature of the spherical output supported wavefronts lay on, will be prescribed, as well as the input supporting and supported wavefronts. This description is adequate to model an optical system with a small aperture stop. A schematic drawing of the system is shown in Fig. 1.

 

Fig. 1 Supporting and supported wavefronts of an optical system with prescribed object and image surfaces. The black rays are perpendicular to both the supported and supporting wavefronts. The red rays are perpendicular only to the supported wavefronts.

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These design conditions lead to a group of three implicit ordinary differential equations in the general case as shown in the next section.

3. Differential equations for double optical surface profiles

We consider a general double surface optical system with refractive index n2 between two surfaces and n1 elsewhere shown in Fig. 2. We will restrict our analysis henceforth to the particular case in which the input supporting wavefront is a sphere centered at the origin O.

 

Fig. 2 Sketch map for a double surface refractive system with prescribed image surface

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(rf (α), α) and (rs(β), β) are points on the first and the second surface respectively, in two polar coordinates with the same origin O. θ 1, θ 2, θ 3 and θ 4 are angles between the ray vector and the surface normal. The prescribed image surface is given in implicit form by S(x, y) = 0. The two surface profiles are defined by the three unknown functions rf (α), rs(β) and β(α).

According to the generalized ray tracing equations [13], the change of the curvatures of the supported wavefronts due to the refractions are expressed by the following equations:

n2cos2θ2ρq2=n1cos2θ1ρq1+(n2cosθ2n1cosθ1)ρqf,n2ρp2=n1ρp1+(n2cosθ2n1cosθ1)ρpf,
n1cos2θ4ρq4=n2cos2θ3ρq3+(n1cosθ4n2cosθ3)ρqs,n1ρp4=n2ρp3+(n1cosθ4n2cosθ3)ρps,
where the suffix q and p refer respectively to the incidence plane and its perpendicular plane (both contain the normal to the surface); ρ together with the suffix f and s refers to the radii of curvature of the first and the second optical surface respectively; ρ together with the suffix 1 and 2 refers to the radii of curvature of the wavefronts before and after the first refraction; ρ together with the suffix 3 and 4 refers to the radii of curvature of the wavefronts before and after the second refraction. The sign convention for the radius of curvature in this paper follows the one stated by Smith [14].

The radii of curvature of the wavefronts between refractions are expressed by:

ρq2D=ρq3andρp2D=ρp3,
where D is the distance between the two points of refraction:

D=(rfsinαrssinβ)2+(rfcosαrscosβ)2.

Anastigmatic designs require ρq4 = ρp4. Then the supported wavefronts are locally spherical after the second refraction. Henceforth we shall restrict the analysis to the case where the second order approximation of the input supported wavefronts are spheres with radius R(α) at the point of the first refraction. So

ρp1=ρq1=R(α).
With Eqs. (1)-(3), (5) and the anastigmatic condition (ρq4 = ρp4) we can eliminate the 8 variables ρqi, ρpi (i = 1, 2, 3, 4) to get a single equation:

(n2(n1R+(n2cosθ2n1cosθ1)ρpf)n2D(n1R+(n2cosθ2n1cosθ1)ρpf)+(n1cosθ4n2cosθ3)ρps)cos2θ4=n2cos2θ3(n1cos2θ1R+(n2cosθ2n1cosθ1)ρqf)n2cos2θ2D(n1cos2θ1R+(n2cosθ2n1cosθ1)ρqf)+n1cosθ4n2cosθ3ρqs.

The loci of the curvature centers of the output supported wavefronts are calculated from the radii of curvature and the direction of the output supported wavefronts. Since the curvature centers have to be on the prescribed image surface, the output supported wavefronts and the prescribed image surface are connected by:

ρp4cos(βatan(drs/dβ)+θ4)+rscosβ=x,ρp4sin(βatan(drs/dβ)+θ4)+rssinβ=y.

Substituting these equations into S(x, y) = 0, we get

S(ρp4cos(βatan(drsdβ)+θ4)+rscosβ,ρp4sin(βatan(drsdβ)+θ4)+rssinβ)=0.

Now get the expression of ρp4 from the second equations in Eqs. (1)-(3) as

ρp4=n1ρpsn2ρps/(n2ρpfρp1n1ρpf+(n2cosθ2n1cosθ1)ρp1D)+(n1cosθ4n2cosθ3).

Finally, let’s derive the following equation from the sine law applied to the triangle formed by the origin and the two points on the two surfaces where the refractions occur:

rs/sin(π(θ1θ2))=rf/sin(βα+θ1θ2).

Consider now Eq. (6) after substituting D (Eq. (4)) into it; Eq. (8) after substituting ρp4 (Eq. (9)) and D (Eq. (4)) into it; and Eq. (10). These three equations relate the variables n1, n2, R, α, β, rf, rs, ρpf, ρqf, ρps, ρqs, θ 1, θ 2, θ 3, θ 4 and the function S(x, y) = 0. ρpf, ρqf, ρps and ρqs can be expressed as functions of α, β, and rf´, rf´´, rs´, β´ (where ′ = d/dα and ″ = d2/dα 2) using the general formulas for rotational surfaces [15] as:

ρpf=rfsinαsin(αatan(rf/rf)),ρqf=(rf2+rf2)3/2rfrfrf22rf2,ρps=rssinβsin(βatan(rs/(rsβ))),ρqs=(rs2+(rs/β)2)3/2rs(rs/β)/βrs22(rs/β)2,
θ 1 and θ 3 can also be written as functions of α, β, rf, rs and β´, rf´, rs´
θ1=atan(rf'rf),θ3=αatan(rf'rf)+asin(n1n2sin(atan(rf'rf)))β+atan(rs'rsβ').
θ 2 and θ 4 can be obtained by Snell’s law from θ 1 and θ 3. Then θ 1, θ 2, θ 3 and θ 4 can also be written as functions of α, β, rf, rs and β´, rf´, rs´.

Since n1, n2, R(α) and the function S(x, y) = 0 is known, the three equations Eqs. (6), (8) and (10) are in fact three implicit ordinary differential equations with three unknown functions rf(α), rs(β) and β(α). The numerical solutions of these functions represent aspheric profiles.

4. Optical design example and evaluation

A design example has been derived using the approach presented in section 3.

For this example R(α) = ∞. The vertexes of the first and the second optical surface are on the axis of the rotational symmetry, 31.5 mm and 41.5 mm away from the origin respectively; the image surface is a flat plane perpendicular to the optical axis and is 85 mm away from the origin (S(x, y)≡x-85mm = 0), so that Eq. (8) becomes

ρp4cos(βatan(drs/dβ)+θ4)+rscosβ=85mm;
the refractive indexes are n1 = 1, n2 = 1.49 and the lens diameter is 42.4 mm. The design can be obtained by solving the three implicit ordinary differential equations using the ode15i function of Matlab. The function requires values of unknown functions and their derivatives at the initial point as boundary conditions. In this example, they are rf, rf´, rf´´, β, β´, rs, rs´/β´ and (rs´/β´)´ at α = α0. To avoid singular point on the symmetrical axis, α0 is chosen as a small increment from 0° so as to maintain the prescribed surface positions. Thus rf = 31.5, rs = 41.5. When rf´ is chosen, β can be obtained by substituting rf, rs and rf´ into Eq. (10). Then rs´/β´ can be obtained by solving Eq. (8). Finally rf´´, β and (rs´/β´)´ can be obtained by solving Eq. (6), differentials of Eqs. (8) and (10) simultaneously. As we can see, the initial value of rf´ will affect the boundary condition which leads to different numerical solutions. A series value of rf´ has been tested and a design in which the two surfaces converge to the same edge without total internal reflection is obtained and shown in Fig. 3(a).

 

Fig. 3 Layouts of double surface refractive anastigmatic design with flat image plane (a), aplanatic with flat image plane (b), optimized anastigmatic design (c) and the optimized aplanatic design (d).

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Once the design has been done, the two image surfaces where the centers of curvature of the tangential and sagittal output supported wavefronts lay on respectively are calculated separately from the surface profiles to check that the design is free from astigmatism. The difference between the two image surfaces and the image plane x = 85 mm is less than 1 μm.

An aplanatic design with the same positions of lens surface vertexes and image plane (Fig. 3(b)) has been used as comparison. Since this design is not anastigmatic there will be in general two surfaces where the centers of curvature of output wavefronts lie and these two surfaces are in general curved unlike the flat single image plane of the anastigmatic design.

All the optical surfaces are fitted and represented by Qcon asphere polynomials [8] in CodeV. A subsequent optimization has been done with a 2mm diameter round pupil centered at the origin in CodeV for 25 field points uniformly distributed in the interval 0° ≤ α ≤ 29.6°. The layouts of the both designs after optimization are plotted in Fig. 3(c) and (d) respectively.

The optimization and the following evaluation have been done for a monochromatic light of wavelength 580 nm. The RMS spot diameter distribution is shown in Fig. 4. The result optimized from anastigmatic design yields an average RMS spot diameter of 9.9 μm and a maximum value of 12.9 μm across 25 field points, while the result optimized from aplanatic design yields an average value of 13 μm and a maximum value of 19 μm.

 

Fig. 4 RMS spot diameter of the anastigmatic and aplanatic designs before and after optimization (left). Modulation Transfer Function across 25 field points of the optimized anastigmatic design (middle) and the optimized aplanatic design (right)

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The Modulation Transfer Function has been evaluated across all the 25 field points and the result is shown in Fig. 4 for both optimized designs. The result optimized from anastigmatic design has achieved an average MTF of 46.9% and a minimum of 42.8% at the frequency of 30 cycles/mm across 25 field points, while the result optimized from aplanatic design achieves an average MTF of 44.5% and a minimum of 32.3% at the same frequency.

5. Discussion and conclusion

The differential equation design approach shown hereinabove can control the tangential and sagittal ray propagation simultaneously, thus anastigmatic design can be readily obtained. We have extended this approach from single optical surface designs to double aspheric surface designs. With more freedoms provided by the second optical surface, both the object and the image surfaces can be prescribed. Nevertheless, in no case, nor single nor double surface design, the mapping from the object to image can be prescribed. This mapping is obtained after the design of the optical surfaces.

Based on this approach, a double aspheric surface anastigmatic design example has been developed. As a good initial design for optimization, a result from the final design with an average RMS spot diameter of 9.9 μm and an average MTF value of 46.9% at the frequency of 30 cycs/mm has been achieved.

Acknowledgment

Authors thank the European Commission (SMETHODS: FP7-ICT-2009-7 Grant Agreement No. 288526, NGCPV: FP7-ENERGY.2011.1.1 Grant Agreement No. 283798), the Spanish Ministries (ENGINEERING METAMATERIALS: CSD2008-00066, SEM: TSI-020302-2010-65 SUPERRESOLUCION: TEC2011-24019, SIGMAMODULOS: IPT-2011-1441-920000, PMEL: IPT-2011-1212-920000), UPM (Q090935C59) and the academic licence for CodeV from Synopsys for the support given to the research activity of the UPM-Optics Engineering Group, making the present work possible.

References and links

1. B. Schmidt, “Ein lichtstarkes komafreies Spiegelsystem,” Zent.-, Ztg. Opt. Mech. 52(2), 25–26 (1931).

2. K. Schwarzschild, “Astronomische Mitteilungen der Königlichen Sternwarte zu Göttingen,” Reprinted: Selected Papers on Astronomical Optics Spie Milestone Ser. 73(3), 1–2 (1993).

3. G. D. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949). [CrossRef]  

4. D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002). [CrossRef]  

5. R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003). [CrossRef]  

6. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005), Chap. 9.

7. J. M. Infante, “Optical Systems Design using SMS method and optimizations,” Ph.D. Thesis, Universidad Politécnica de Madrid (2013).

8. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007). [CrossRef]   [PubMed]  

9. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express 18(19), 19700–19712 (2010). [CrossRef]   [PubMed]  

10. Y. Nie, F. Duerr, and H. Thienpont, “Direct design approach to calculate a two-surface lens with an entrance pupil for application in wide field-of-view imaging,” Opt. Eng. 54(1), 015102 (2015). [CrossRef]  

11. J. Liu, P. Benítez, and J. C. Miñano, “Single freeform surface imaging design with unconstrained object to image mapping,” Opt. Express 22(25), 30538–30546 (2014). [CrossRef]   [PubMed]  

12. J. Liu, J. C. Miñano, P. Benítez, and L. Wang, “Single optical surface imaging designs with unconstrained object to image mapping,” Proc. SPIE 8550, 855011 (2012). [CrossRef]  

13. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics: The k-function and its Ramifications (Wiley-VCH, 2006).

14. W. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007), Chap. 3.

15. E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition (Textbooks in Mathematics) (Chapman and Hall/CRC, 2006)

References

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  1. B. Schmidt, “Ein lichtstarkes komafreies Spiegelsystem,” Zent.-, Ztg. Opt. Mech. 52(2), 25–26 (1931).
  2. K. Schwarzschild, “Astronomische Mitteilungen der Königlichen Sternwarte zu Göttingen,” Reprinted: Selected Papers on Astronomical Optics Spie Milestone Ser. 73(3), 1–2 (1993).
  3. G. D. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
    [Crossref]
  4. D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
    [Crossref]
  5. R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
    [Crossref]
  6. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005), Chap. 9.
  7. J. M. Infante, “Optical Systems Design using SMS method and optimizations,” Ph.D. Thesis, Universidad Politécnica de Madrid (2013).
  8. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15(8), 5218–5226 (2007).
    [Crossref] [PubMed]
  9. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Express 18(19), 19700–19712 (2010).
    [Crossref] [PubMed]
  10. Y. Nie, F. Duerr, and H. Thienpont, “Direct design approach to calculate a two-surface lens with an entrance pupil for application in wide field-of-view imaging,” Opt. Eng. 54(1), 015102 (2015).
    [Crossref]
  11. J. Liu, P. Benítez, and J. C. Miñano, “Single freeform surface imaging design with unconstrained object to image mapping,” Opt. Express 22(25), 30538–30546 (2014).
    [Crossref] [PubMed]
  12. J. Liu, J. C. Miñano, P. Benítez, and L. Wang, “Single optical surface imaging designs with unconstrained object to image mapping,” Proc. SPIE 8550, 855011 (2012).
    [Crossref]
  13. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics: The k-function and its Ramifications (Wiley-VCH, 2006).
  14. W. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007), Chap. 3.
  15. E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition (Textbooks in Mathematics) (Chapman and Hall/CRC, 2006)

2015 (1)

Y. Nie, F. Duerr, and H. Thienpont, “Direct design approach to calculate a two-surface lens with an entrance pupil for application in wide field-of-view imaging,” Opt. Eng. 54(1), 015102 (2015).
[Crossref]

2014 (1)

2012 (1)

J. Liu, J. C. Miñano, P. Benítez, and L. Wang, “Single optical surface imaging designs with unconstrained object to image mapping,” Proc. SPIE 8550, 855011 (2012).
[Crossref]

2010 (1)

2007 (1)

2003 (1)

R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[Crossref]

2002 (1)

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
[Crossref]

1993 (1)

K. Schwarzschild, “Astronomische Mitteilungen der Königlichen Sternwarte zu Göttingen,” Reprinted: Selected Papers on Astronomical Optics Spie Milestone Ser. 73(3), 1–2 (1993).

1949 (1)

G. D. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

1931 (1)

B. Schmidt, “Ein lichtstarkes komafreies Spiegelsystem,” Zent.-, Ztg. Opt. Mech. 52(2), 25–26 (1931).

Benítez, P.

J. Liu, P. Benítez, and J. C. Miñano, “Single freeform surface imaging design with unconstrained object to image mapping,” Opt. Express 22(25), 30538–30546 (2014).
[Crossref] [PubMed]

J. Liu, J. C. Miñano, P. Benítez, and L. Wang, “Single optical surface imaging designs with unconstrained object to image mapping,” Proc. SPIE 8550, 855011 (2012).
[Crossref]

Duerr, F.

Y. Nie, F. Duerr, and H. Thienpont, “Direct design approach to calculate a two-surface lens with an entrance pupil for application in wide field-of-view imaging,” Opt. Eng. 54(1), 015102 (2015).
[Crossref]

Forbes, G. W.

Liu, J.

J. Liu, P. Benítez, and J. C. Miñano, “Single freeform surface imaging design with unconstrained object to image mapping,” Opt. Express 22(25), 30538–30546 (2014).
[Crossref] [PubMed]

J. Liu, J. C. Miñano, P. Benítez, and L. Wang, “Single optical surface imaging designs with unconstrained object to image mapping,” Proc. SPIE 8550, 855011 (2012).
[Crossref]

Lynden-Bell, D.

R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[Crossref]

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
[Crossref]

Miñano, J. C.

J. Liu, P. Benítez, and J. C. Miñano, “Single freeform surface imaging design with unconstrained object to image mapping,” Opt. Express 22(25), 30538–30546 (2014).
[Crossref] [PubMed]

J. Liu, J. C. Miñano, P. Benítez, and L. Wang, “Single optical surface imaging designs with unconstrained object to image mapping,” Proc. SPIE 8550, 855011 (2012).
[Crossref]

Nie, Y.

Y. Nie, F. Duerr, and H. Thienpont, “Direct design approach to calculate a two-surface lens with an entrance pupil for application in wide field-of-view imaging,” Opt. Eng. 54(1), 015102 (2015).
[Crossref]

Schmidt, B.

B. Schmidt, “Ein lichtstarkes komafreies Spiegelsystem,” Zent.-, Ztg. Opt. Mech. 52(2), 25–26 (1931).

Schwarzschild, K.

K. Schwarzschild, “Astronomische Mitteilungen der Königlichen Sternwarte zu Göttingen,” Reprinted: Selected Papers on Astronomical Optics Spie Milestone Ser. 73(3), 1–2 (1993).

Thienpont, H.

Y. Nie, F. Duerr, and H. Thienpont, “Direct design approach to calculate a two-surface lens with an entrance pupil for application in wide field-of-view imaging,” Opt. Eng. 54(1), 015102 (2015).
[Crossref]

Wang, L.

J. Liu, J. C. Miñano, P. Benítez, and L. Wang, “Single optical surface imaging designs with unconstrained object to image mapping,” Proc. SPIE 8550, 855011 (2012).
[Crossref]

Wassermann, G. D.

G. D. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

Willstrop, R. V.

R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[Crossref]

Wolf, E.

G. D. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

Mon. Not. R. Astron. Soc. (2)

D. Lynden-Bell, “Exact optics: a unification of optical telescope design,” Mon. Not. R. Astron. Soc. 334(4), 787–796 (2002).
[Crossref]

R. V. Willstrop and D. Lynden-Bell, “Exact optics – II. Exploration of designs on- and off-axis,” Mon. Not. R. Astron. Soc. 342(1), 33–49 (2003).
[Crossref]

Opt. Eng. (1)

Y. Nie, F. Duerr, and H. Thienpont, “Direct design approach to calculate a two-surface lens with an entrance pupil for application in wide field-of-view imaging,” Opt. Eng. 54(1), 015102 (2015).
[Crossref]

Opt. Express (3)

Proc. Phys. Soc. B (1)

G. D. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B 62(1), 2–8 (1949).
[Crossref]

Proc. SPIE (1)

J. Liu, J. C. Miñano, P. Benítez, and L. Wang, “Single optical surface imaging designs with unconstrained object to image mapping,” Proc. SPIE 8550, 855011 (2012).
[Crossref]

Reprinted: Selected Papers on Astronomical Optics Spie Milestone Ser. (1)

K. Schwarzschild, “Astronomische Mitteilungen der Königlichen Sternwarte zu Göttingen,” Reprinted: Selected Papers on Astronomical Optics Spie Milestone Ser. 73(3), 1–2 (1993).

Zent.-, Ztg. Opt. Mech. (1)

B. Schmidt, “Ein lichtstarkes komafreies Spiegelsystem,” Zent.-, Ztg. Opt. Mech. 52(2), 25–26 (1931).

Other (5)

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics (Academic, 2005), Chap. 9.

J. M. Infante, “Optical Systems Design using SMS method and optimizations,” Ph.D. Thesis, Universidad Politécnica de Madrid (2013).

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics: The k-function and its Ramifications (Wiley-VCH, 2006).

W. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007), Chap. 3.

E. Abbena, S. Salamon, and A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition (Textbooks in Mathematics) (Chapman and Hall/CRC, 2006)

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Figures (4)

Fig. 1
Fig. 1 Supporting and supported wavefronts of an optical system with prescribed object and image surfaces. The black rays are perpendicular to both the supported and supporting wavefronts. The red rays are perpendicular only to the supported wavefronts.
Fig. 2
Fig. 2 Sketch map for a double surface refractive system with prescribed image surface
Fig. 3
Fig. 3 Layouts of double surface refractive anastigmatic design with flat image plane (a), aplanatic with flat image plane (b), optimized anastigmatic design (c) and the optimized aplanatic design (d).
Fig. 4
Fig. 4 RMS spot diameter of the anastigmatic and aplanatic designs before and after optimization (left). Modulation Transfer Function across 25 field points of the optimized anastigmatic design (middle) and the optimized aplanatic design (right)

Equations (13)

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n 2 cos 2 θ 2 ρ q2 = n 1 cos 2 θ 1 ρ q1 + ( n 2 cos θ 2 n 1 cos θ 1 ) ρ qf , n 2 ρ p2 = n 1 ρ p1 + ( n 2 cos θ 2 n 1 cos θ 1 ) ρ pf ,
n 1 cos 2 θ 4 ρ q4 = n 2 cos 2 θ 3 ρ q3 + ( n 1 cos θ 4 n 2 cos θ 3 ) ρ qs , n 1 ρ p4 = n 2 ρ p3 + ( n 1 cos θ 4 n 2 cos θ 3 ) ρ ps ,
ρ q2 D= ρ q3 and ρ p2 D= ρ p3 ,
D= ( r f sinα r s sinβ ) 2 + ( r f cosα r s cosβ ) 2 .
ρ p1 = ρ q1 =R(α).
( n 2 ( n 1 R + ( n 2 cos θ 2 n 1 cos θ 1 ) ρ pf ) n 2 D( n 1 R + ( n 2 cos θ 2 n 1 cos θ 1 ) ρ pf ) + ( n 1 cos θ 4 n 2 cos θ 3 ) ρ ps ) cos 2 θ 4 = n 2 cos 2 θ 3 ( n 1 cos 2 θ 1 R + ( n 2 cos θ 2 n 1 cos θ 1 ) ρ qf ) n 2 cos 2 θ 2 D( n 1 cos 2 θ 1 R + ( n 2 cos θ 2 n 1 cos θ 1 ) ρ qf ) + n 1 cos θ 4 n 2 cos θ 3 ρ qs .
ρ p4 cos( βatan( d r s / dβ )+ θ 4 )+ r s cosβ=x, ρ p4 sin( βatan( d r s / dβ )+ θ 4 )+ r s sinβ=y.
S( ρ p4 cos( βatan( d r s dβ )+ θ 4 )+ r s cosβ, ρ p4 sin( βatan( d r s dβ )+ θ 4 )+ r s sinβ )=0.
ρ p4 = n 1 ρ ps n 2 ρ ps / ( n 2 ρ pf ρ p1 n 1 ρ pf +( n 2 cos θ 2 n 1 cos θ 1 ) ρ p1 D ) +( n 1 cos θ 4 n 2 cos θ 3 ) .
r s / sin( π( θ 1 θ 2 ) ) = r f / sin( βα+ θ 1 θ 2 ) .
ρ pf = r f sinα sin( αatan( r f / r f ) ) , ρ qf = ( r f 2 + r f 2 ) 3/2 r f r f r f 2 2 r f 2 , ρ ps = r s sinβ sin( βatan( r s / ( r s β ) ) ) , ρ qs = ( r s 2 + ( r s / β ) 2 ) 3/2 r s ( r s / β ) / β r s 2 2 ( r s / β ) 2 ,
θ 1 =atan( r f ' r f ), θ 3 =αatan( r f ' r f )+asin( n 1 n 2 sin( atan( r f ' r f ) ) )β+atan( r s ' r s β' ).
ρ p4 cos( βatan( d r s / dβ )+ θ 4 )+ r s cosβ=85mm;

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