## Abstract

This paper describes a method to design families of singlet lenses free of all orders of spherical aberration. These lenses can be mass produced according to Schwarzschild's formula and therefore one can find many practical applications. The main feature of this work is the application of an analysis that can be extended to grazing or maximum incidence on the first surface. Also, here, the authors present some developments that corroborate geometrical optics results, along with the axial thick lensmaker's formula, which can be applicable to any pair of finite conjugate planes for any lens shape (bending) and can be used instead of the classical thick lensmaker's formula, which always assumes that the object is at infinity, to attain better accuracy.

## 1. Introduction

Aspheric lenses are designed and made to reduce optical system aberrations. Of all the aberrations a lens may have, the most studied and worked out is the spherical aberration. The first studied were lenses with refractive surfaces free of all orders of spherical aberration corresponding to non-degenerated Cartesian ovals of revolution, or degenerated, as ellipsoids, or hyperboloids of revolution. These were described by Descartes [1], and have recently been described with explicit examples by Valencia-Estrada *et al.* [2]. Subsequently, Luneberg [3] established a method for calculating the geometry of the second surface from an initial first surface that introduces spherical aberration, but he only described two particular cases and did not complete his notes. An aplanatic system can be designed using aspheric surfaces, as pointed out by Wassermann & Wolf [4] and also as described in the book by Born *et al.* [5].

Wasserman & Wolf proposed to use two aspheric adjacent surfaces to correct spherical and coma aberrations, with a solution consisting of two first-order simultaneous differential equations, which are solved numerically according to Malacara-Hernández *et al.* [6].

The spherical aberration has been studied by several researchers, trying to get the best image on the axis with accurate models [7] or using high-order polynomial approximations ([8,9], and outstanding work done by Buchdahl [10,11]).

Also, many specialized optical systems have been developed to greatly reduce the spherical aberration, but none has eliminated it completely. Designs with single or multiple lenses were developed [12–23].

With the invention of optical discs for digital recording, hundreds of designs became available by Visser *et al.* [23], Stallinga [24] and Frolov *et al.* [25] for the optical pick-up head, with a significant reduction of spherical aberration, allowing the disc capacity to be increased with technologies like CD, DVD, HD DVD and Blu-ray.

With the advent of intraocular lenses (IOL) to replace the lens in the human eye, multiple monofocal or multifocal designs [26] have appeared on the market, with reduced or controlled spherical aberration.

Also, recently there have appeared numerous studies on the design of single aspherical lenses [27,28], trying to find the best corrective surfaces with an approach that fulfils Schwarzschild's formula. Among them the work performed by Avendaño-Alejo [29] and Kiontke *et al.* stand out [30], describing a corrective surface with an approximate representation featuring a hyperbolic meridional section according to Valencia-Estrada [2], when the object is at infinity and the other surface is flat. Aspheric deformation coefficients are described as functions of the conic constant and the vertex curvature.

More precision in manufacturing has also been achieved, reaching an RMS roughness and formal errors of about less than 10 (nm) [30]; it has also been possible to further reduce spherical aberration, with the design of optical systems using multiple lenses, for photolithography [31]. This is why the dimensions of design tolerances are given with a precision of 1 (nm) in all figures presented here, almost reaching the limit of interatomic distances.

Finally, no one else has studied, accurately and analytically, how the shape of the second surface of a lens is free of spherical aberration after refraction at a first smooth rotational symmetric interface of any kind, since, in many applications, it is practical to completely correct the spherical aberration aspherizing a single surface. In this paper, novel solutions are presented, which rigorously calculate, in cylindrical coordinates, the geometry of the second surface *z*_{b}(*r*) for any given first surface preset *z*_{a}(*r*), through an original method.

## 2. Physical–mathematical model

A lens is a translucent body with a relative refraction index *n*, consisting of one or more refractive surfaces. A simple lens is basically composed of two surfaces: an anterior one, through which light enters, and a posterior one, from which light emerges. Using different coordinate systems, the geometry of these surfaces can be mathematically characterized in many ways, which can be explicit, parametric or implicit.

Common lenses (singlets) are made of two surfaces of revolution with a common axis, called the optical axis, which also corresponds to rotationally symmetrical lenses described in a cylindrical coordinate system (*r*,*z*). In this document, the anterior and the posterior surfaces are designated with subscripts *a* and *b* respectively, establishing an absolute coordinate origin at the point of intersection of the optical axis with the first surface (first surface vertex). Thus, the lens surfaces are explicitly defined by functions *z*_{a}(*r*) and *z*_{b}(*r*), where *z* is the sagitta measured parallel to the optical axis, from the tangent plane at the first surface vertex, and *r* is the cylindrical radius or the height to the optical axis, at which the sagitta is measured.

Rotationally symmetrical lens surfaces can be flat, spherical or aspherical. Flat surfaces are quite common and are specified in cylindrical coordinates by a translational constant *k* as *z*=*k*. Spherical surfaces are the most common and can be specified in many ways; the representation used here, in cylindrical coordinates, is *R* is the radius of the sphere, and *k* is the translational constant from the origin. The most common aspherical surfaces are specified with respect to a conic reference surface according to Schwarzschild's formula as
*c*=1/*R* is the central curvature, *K* is a conic constant and *C*_{2j} are additional even coefficients of deformation.

The mathematical and physical models normally used in geometrical optics perform many approximations, i.e. paraxial theory, but they are far from being mathematically accurate.

A lens featuring simple spherical surfaces leads to the well-known ‘spherical aberration’ which prevents obtaining a stigmatic image of a point-object located on the optical axis.

A physical–mathematical model that obeys rigorous geometrical optics is presented here for ray tracing by means of calculus and vector analysis in two dimensions, since rotationally symmetrical lenses are assumed. This ray tracing model applies to a single lens, with axial thickness *t*, a prescribed anterior spherical surface *z*_{a}(*r*), and an unknown posterior *z*_{b}(*r*) surface.

The ideal optical system of such a lens has an object-point on the optical axis with an anterior vertex-object distance *t*_{a}, positive defined in the positive direction of the optical axis *z*. From such point departs a ray that reaches an arbitrary point on the first surface, whose coordinates are (*r*_{a},*z*_{a}(*r*_{a})), where it is refracted by the lens due to its relative refraction index *n*>1, changing its propagation direction according to Snell's Law. This ray arrives at the back surface at a point on the second surface (*r*_{b},*z*_{b}(*r*_{b})), where the ray is again refracted according to Snell's Law to reach a pre-established image point on the optical axis. This image point is located at a back vertex to image distance *t*_{b}, positively defined in the positive direction of the optical axis *z* (figure 1).

The purpose of this study was to determine the mathematical prescription of the posterior surface *z*_{b}(*r*), so that the optical system, as shown in figure 1, with any previously defined anterior surface *z*_{a}(*r*), and known parameters *n*, *t*_{a}, *t*_{b} and *t*, to be free of all spherical aberration orders.

This problem has been analytically studied by some authors such as Luneberg [3] and Born & Wolf [5] with principles that can be applied to an optical system, as shown in figure 1. It is worth mentioning that Luneburg modelled the problem for two cases: when the first surface is flat with a finite object-distance and the image towards infinity, and when the first surface is a convex spherical with the object at infinity and a finite image-distance; however, his analysis was not extended to other cases.

Considering spherical or flat incoming wavefronts, and using Fermat's Principle, for a spherical aberration-free lens, the optical path of any non-central ray (figure 1) must be equal to the optical path of the axial ray, which travels along the optical axis; therefore,

Snell's Law at the first surface provides the second fundamental equation required to solve the problem
*n* is the refraction index of the lens relative to the surrounding environment, *θ*_{i} is the angle of incidence of the ray with respect to the unit normal vector *θ*_{r} is the angle of the refracted ray with respect to the unit normal vector

The constructed unit vectors correspond to the row vectors

Replacing the unitary vectors equation (2.4) in equation (2.3) a fundamental equation is obtained

From equations (2.2) and (2.5), the general solution can be deduced, which is the rear surface's geometry that corrects the spherical aberration, for a first prescribed planar or non-planar surface. The object can be at infinity and the image at a finite distance, or both the object and image are at finite distances, for any prescribed smooth (mathematically) rotationally symmetrical anterior surface.

The parametric general solution for *r*_{b} is odd:
*z*_{b} is even and meets
*s*_{1−5}, which can be sign functions of the abscissa *r*_{a}, object distance *t*_{a}, image distance *t*_{b} or some special function. The solution equations (2.6)–(2.8) are valid for isotropic and homogeneous materials or metamaterials with negative refraction index, for a valid aperture and in the spherical case, for all *t*_{a}≠*R*_{a}, with recurrent and parametric variables {*A*-*R*}:

Since the parametric function *r*_{b} is an odd function of *r*_{a}, with an inflection point at *r*_{a}=0, then *r*^{′}(*r*_{a}=0) exists and *r*^{′′}(*r*_{a}=0)=0. Moreover, since *z*_{b} is an even function of *r*_{a} with either a local maximum or minimum at *r*_{a}=0, then *r*_{a}=0, we have

If the general solution for *r* according to equation (2.6) is factorizable by *r*_{a}, then *r* can be written in the form *r*=*r*_{a}*m*, where *m* is an even function of *r*_{a}. Thus, the first derivative of *r* under the chain rule is equivalent to

If the derivative of *m* with respect to *r*_{a} is an odd function of *r*_{a}, and if the vertex is a regular point (equivalent to say that if *m* is expandable in an even power series), its derivative can be factorizable by *r*_{a} in the numerator. So the *r*_{a} that premultiplied the derivative of *m* does not vanish; thus, the limit of the first summand vanishes mandatorily. Then, evaluating the curvature at the limit when *r*_{a}→0, we have

Therefore, the vertex curvature of any second surface correcting the spherical aberration is reduced to

## 3. An example

In the following figures, some results obtained for a front convex spherical surface (*R*_{a}>0) are shown, whose sagitta is given by

### (a) Thick lens solutions without inversion of internal rays, with object point at infinity and image at a finite distance

Evaluating equations (2.6)–(2.8) with the recurrent variables according to equation (2.9) for equation (3.1), the simplified general solution, for a first convex spherical surface with an object at infinite distance (using new recurring lowercase variables *a*,…,*m*), is

#### (i) With a finite real image

To verify equations (3.1)–(3.5) ray tracing was performed for hundreds of designs. Figure 2 shows two examples of the attainable solutions (rays propagate from the bottom up) evaluating equations (3.3)–(3.5). Also, to verify the solution, corroborating the paraxial theory, the posterior vertex curvature of these sphero-aspheric lenses, with an object at infinity, and a real image at *t*_{b} behind the lens, is obtained evaluating equation (2.13) with equations (3.2)–(3.5):
*R*_{b}

Rewriting equation (3.7) similar to the classical Gaussian form
*F*_{b}=*t*_{b} for thick lenses that is described by Malacara-Hernández [6], and so
*f* corresponds to the focal distance measured from the principal plane, as defined for an object located at infinity.

### (b) Thick lens solutions without internal ray inversion, with object and image at finite real or virtual distances

Evaluating equations (2.6)–(2.8) with the recurrent variables according to equation (2.9) for equation (3.1), the simplified general solution, for a first convex spherical surface with a finite object distance, is ∀*R*_{a}>0, *t*_{a}≠*R*_{a} and *t*_{b}≠0:

*R*_{a}>0 and *t*_{a}=*R*_{a}, we have

#### (i) With a real object and a finite real image

In this case, to verify equation (3.1) and equations (3.10)–(3.13) with the rules established in equations (3.15)–(3.16), ray tracing was performed for hundreds of configurations. Figure 4 shows three examples of the solutions found when rays travel upwards. Also, to verify the solution, corroborating the paraxial theory, the posterior vertex curvature of these sphero-aspheric lenses with a proximal or distant real object and a real image is obtained by means of the pinching theorem described by Weisstein [33], at the points of discontinuity when *t*_{a}=−*R*_{a}/(*n*−1) and *t*_{a}=−*R*_{a}/(*n*^{2}−1):
^{1} ^{,}^{2}
^{1,2}
*R*_{b} for thick lenses, which can also be compared with the best approach, the Gauss–Gullstrand formula:

Figure 5*b*–*d* shows the ray tracing for the best bi-spherical approximations to the zero spherical aberration solution shown in figure 5*a*.

#### (ii) With a real object and a finite virtual image

Solving this case, to verify equation (3.1) and equations (3.10)–(3.13) with the set rules of equations (3.15) and (3.16) ray tracing was once more performed for hundreds of designs. In figure 6, two typical solutions are shown.

#### (iii) With a virtual object and a finite real image

To verify equation (3.1) and equations (3.10)–(3.14) with the rules of equations (3.15) and (3.16) ray tracing was performed for hundreds of designs. In figure 7, two of the solutions found are illustrated.

## 4. Axial thick lensmaker's formulae

The central back curvature of these sphero-aspherical lenses also satisfies the mnemonic formula if *t*_{a}>*R*_{a}, but if *t*_{a}=*R*_{a}, another formula may be used (piecewise solution): this is an example for a first convex spherical surface with finite conjugate planes.

It has been checked above, according to paraxial theory, that the posterior vertex curvature of these sphero-aspherical lenses, with a convex spherical first surface and a close object, is obtained by evaluating the general solution according to equations (3.10)–(3.13) when *t*_{a}≠*R*_{a}, and with the particular solution equation (3.14) when *t*_{a}=*R*_{a}. Thus, three cases arise, depending on the domain of *t*_{a}, after using the pinching theorem described by Weisstein [32] at the five points of discontinuity of the apical curvature (irregular points): if the object is real when *t*_{a}=−*R*_{a}/(*n*−1) and *t*_{a}=−*R*_{a}/(*n*^{2}−1), or virtual when *t*_{a}=*R*_{a}/(*n*^{2}+1), *t*_{a}=*R*_{a}/(*n*+1) and *t*_{a}=*R*_{a}:

### (a) First case: − ∞ < t a < R a

The vertex curvature is given by equation (3.17) corresponding to a relative or Gaussian central curvature radius equation (3.18), which also corresponds to the mnemonic formula equation (3.19).^{1,2}

### (b) Second case: *t*_{a}=*R*_{a}

*t*_{a}→*R*_{a}, and corresponds to the Gaussian radius

### (c) Third case: *t*_{a}>*R*_{a}

The vertex curvature equation (3.17) corresponding to a relative or Gaussian central curvature radius equation (3.18), which also corresponds to the mnemonic formula equation (3.19),^{1,2} whose limits, when *t*_{a}→*R*_{a}, converge to the same results of the above second case. It is easy to demonstrate that the limits of the mnemonic equation (3.19) when the central thickness *t* approaches zero converge in the paraxial thin lensmaker's formula. It is also possible to find the paraxial thick lensmaker's formula, computing the limit when

It is also important to establish the conditions to prevent quadrant inversion of the rays while propagating through the lens. To determine this condition, Snell's Law can be used at the first interface, evaluated when *r*_{b}=0 and *z*_{b}=*t*, while always *t*<*R*_{a}
*t*_{ac}>0 and *t*_{ac}<*R*_{a}:
*t*_{b}<0 and

## 5. Solutions with Schwarzschild's formula

There exist an infinite number of aspherical approximated surfaces, according to Schwarzschild's formula, that permit the minimizing of all orders of spherical aberration, generated by one spherical surface with radius *R*_{a}. The simplest solution may be represented by the canonical form
*K*, and a reduced number *N* of relevant aspheric coefficients *C*_{2j} according to the recommendations by Forbes [33].

The exact solution equation (5.1) can be expanded as even powers series around its vertex, using the Z inverse transform of the conic summand with *k*=2*j*:

To calculate the best fitted conic constant *K* and *N* aspheric coefficients *C*_{2j}, it is recommended to follow one procedure: statistical or analytical. The results may or may not be dependent of the aperture diameter *d*_{l}.

### (a) Statistical methods

The most commonly used expression is the polynomial least-squares fitting procedure

The best fit can be obtained by solving the nonlinear system equation (5.5) with *N*+1 unknowns: *K* and *N* coefficients (*C*_{2j})

The data for the ‘experimental’ *p*-points with coordinates (*r*_{a}(*p*),*z*_{b}(*r*_{b}(*r*_{a}(*p*)))) in the interval 0<*r*_{b}(*r*_{a}(*p*))<*d*_{l}/2 may be calculated using the recurrent exact solutions based on equations (2.6)–(2.9). At the end, the best-fit result depends on the value of the correlation coefficient according to Bates & Watts [34]. But this approach is time-consuming and the mental effort will be greater than by using the exact recurrent solution for fine interpolation with an expert CAM, with or without tool correction as is recommended by Weck [35] and Valencia-Estrada *et al.* [38], to generate ISO codes (G codes) to be executed in computerized numerical control (CNC) machines. The fine interpolation described by Weck [35] refers to one of the available methods to segment the trajectory of a tool in a CNC machine. For the machining of a lens it is always better to make a single stage fine segmentation of the tool path relative to the workpiece, since it generates better surface quality and better form, rather than a multi-stage algorithm with a first rough stage. A programmer with basic programming skills can develop all the needed algorithms to create an expert system for fine interpolation of these corrective surfaces (CAM), pre-setting a very small sagittal error for each segment, which depends on the resolution of the feed axes and of the measuring system: i.e. 1 (nm) at current technological achievements. In many instances, current lens design optimization programmes can find the form of the correcting surface without too much difficulty, but the required algorithms to perform the tool offset in the CNC machines are longer and can take a lot of computation time as the number of coefficients increases to achieve the 1 nm precision. Furthermore, large aperture designs with low working *F*#, may need a large number of coefficients, and at the end can present an undesired and considerable residual amount of spherical aberration, since for steep profiles a 1 nm machining precision might not be enough to reach the mathematical atomic-level description of the desired surface.

### (b) Analytical methods

One solution can be obtained if *z*_{b} can be expanded in even power series around its vertex, as
*j*=2,3,…,*N*
*j* and solving for the unknown aspherical coefficients *C*_{2j}:

As a proof, coefficient *A*_{2} must correspond to the second derivate of the even parametric function *z*_{b}(*r*_{b}(*r*_{a})) with respect to *r*_{b}, evaluated around its vertex, divided by 2!

All coefficients *A*_{2j} can be calculated using the chain rule to implicitly derive the parametric functions *z*_{b} and *r*_{b}. So, in general, performing operations with the same principles to obtain equation (5.10): using the chain rule, implicitly deriving and vanishing: odd *z*_{b} derivatives and even *r*_{b} derivatives; coefficients *A*_{2j} of the power series of *z*_{b} around its vertex, correspond to:
*R*_{k} y *Z*_{k} defined as
*C*_{2j}(*K*) as a function of the conic constant, and therefore there will be infinite solutions. To find a good solution for the conic constant that determines the best fit, there are available two practical cases: with the analytical or numerical solution for *K* in
*K* near zero, in
*r*_{p} is a radius corresponding to the parameter *r*_{p}=*r*_{a} when *r*(*r*_{a})=*d*_{d}/2 that correspond to the back aperture. Once a suitable conic constant has been obtained, this can be iteratively optimized with very small changes of *K* to minimize the spherical aberration. An example of the results following this method, asphericity coefficients according to Schwarzschild's formula (which determine the best fit of the correcting surface of spherical aberration for a first planar interface), obtained by means of equations (2.6) and (2.7) according to (2.9) with signs *s*_{1}=−1, *s*_{2}=−sign(*t*_{a}) sign(*r*_{a}), *s*_{4}=−*s*_{2} and *s*_{5}=sign(*t*_{b})*s*_{4}, are
*U*=*nt*_{a}−*t* and polynomials *P*_{2j} that correspond to

The coefficients calculated *C*_{2j} with a relative refractive index *n*=*n*_{i}/*n*_{a}, in the limit when *t*_{b} tends to infinity, match the coefficients recently released by Castillo-Santiago *et al.* [37].

Figure 9 shows ray tracing for a flat-aspherical lens with reduced spherical aberration, with coefficients calculated according to equation (5.15) with polynomial equation (5.16).

Also, we can obtain the best fit of the Schwarzschild's correcting surface of spherical aberration for a first convex spherical interface, for an object at infinity obtained by means of the rigorous solutions according to equations (3.1)–(3.5), with vertex curvature by equation (3.6) and four aspheric coefficients *C*_{2j} according to equation (5.9)
*U*=*n*(*R*_{a}−*t*)+*t* and coefficients *P*_{2j} that correspond to

It is noteworthy that the size of the polynomials increases as the order increases and the number of variables required to represent the first surface. These expressions are too long to be presented here, but some should be presented in future work.

Figure 10 shows ray tracing for a sphero-aspherical lens with reduced spherical aberration, with coefficients calculated according to equation (5.17) with polynomials equation (5.18).

## 6. Conclusion

The aspheric back surfaces found to correct all orders of spherical aberration, generated by any kind first surface, can be represented according to general rigorous solution equations (2.6)–(2.9), but when the first surface is convex-spherical and the object is far away, equations (2.6)–(2.9) can be reduced to equations (3.2)–(3.5), and when the first surface is convex-spherical and the object is near, can be reduced to equations (3.10)–(3.14) with signs rules of equations (3.15) and (3.16).

Approximated solutions with reduced spherical aberration can be represented with Schwarzschild's formula with deformation coefficients calculated using equations (5.1)–(5.3) and equation (5.9). Two approximated analytical solutions are described: when the first surface is flat, and the object is at a finite distance, with deformation coefficients represented with equations (5.15) and (5.16); and when the first surface is convex-spherical and the object is at infinity, with deformation coefficients represented by equations (5.17) and (5.18). The approximated correcting back surface found may be machined using many kinds of approximate solutions according to standard ISO 10110:(2007), using statistical and analytical methods, but, generally requiring more time and computing resources than using the general solution for fine interpolation in CNC machines.

It is important to note that although these corrective surfaces are developed to design lenses free of spherical aberration, they can have a considerable amount of comatic aberration, or large surface slopes, defects that may mean that solutions are not always practical. However, according to the results presented herein, it is possible to obtain an analytical solution of a first surface so that the second surface can be calculated with the method presented here, ensuring zero spherical aberration and zero meridional coma at the edge of the field. However, the lens will show residual coma at intermediate points, and some oblique and residual coma for all off-axis points, assuming always that rays have inversion within the lens.

Based on experimental results for hundreds of combinations of the input variables, the approximate solution shows that equation (5.1) with reduced spherical aberration performs very well except for some critical cases which need to be verified graphically using ray tracing software when working at very small F-numbers, to ensure that the series converges quickly. Correcting surfaces (back or anterior because designs are reversible, swapping *t*_{a} and *t*_{b} in all equations) proposed here may be made using many kinds of approximate solutions, like Forbes representation [33] not shown here, also using statistical and analytical methods, but, generally taking more time and computing resources than the general solution for fine interpolation in CNC machines.

Also, the proposed aspherical lenses with zero spherical aberration (assuming that all incident radiation is refracted, that the lens material is ideally isotropic and homogeneous and that the refractive interfaces are ideally continuous) will have a Point Spread Function (PSF) at the image point on the optical axis, produced by a wavefront shape that will result as a consequence of which of the following conditions are satisfied or not: the diffraction (with or without apodization), the successive internal reflections and refractions of light that is not radially polarized, the resolution of the surface at an atomic scale, and, the self-phase modulation and from the nonlinear effects of dispersion when luminous intensities are high. Wavefronts can also be refracted in the *z*-axis positive direction, with the wave's electric field always oscillating on the *R*–*Z* planes (radial polarization), thereby yielding an image point spread, which should not be confused with the spherical aberration nor with diffraction effects.

The axial thick lensmaker's formula with finite conjugate object and image planes allow manufacturing lenses in which the nominal point-image coincides with the paraxial point-image (figure 5), which can be verified by computer ray tracing. The formulae are also valid for a first concave spherical surface, but their critical values are always real. Also, it is important to remember that this type of lenses constitutes a little breakthrough [38], so that a new generation of optical instruments can be designed that will allow the progress of many human disciplines, recalling that a portion of the market is composed by single-lens optical systems.

## Data accessibility

This work does not have any experimental data.

## Author contributions

J.C.V.E. developed most of the physical–mathematical model and programming. R.B.F.H. developed the remaining portion of the model and reviewed and verified all results. D.M.H. reviewed and verified all results also. All authors gave final approval for publication.

## Funding statement

We have no funding or grants.

## Conflict of interests

We have no competing interests.

## Acknowledgements

Centro de Investigaciones en Optica A.C. CIO, León, Guanajuato, México, and Consejo Nacional de Ciencia y Tecnología de México, CONACYT, for its economic support.

## Footnotes

↵1 Formulae for the lensmaker with finite conjugate planes, valid for all lens without quadrant inversion within the same (I and II figure 1).

↵2 For the formulae equations (3.10)–(3.13) to give valid results, the anterior vertex-object distance

*t*_{a}, of its five irregular values, should be of at least 1 (nm), either above or below these four critical values, if the object is real when*t*_{a}=−*R*_{a}/(*n*−1) and*t*_{a}=−*R*_{a}/(*n*^{2}−1), or virtual when*t*_{a}=*R*_{a}/(*n*^{2}+1) and*t*_{a}=*R*_{a}/(*n*+1); when*t*_{a}=*R*_{a}, it should be below the critical point.

- Received August 8, 2014.
- Accepted December 22, 2014.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.