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Abstract
An analytically designed aplanatic meniscus lens is proposed for increasing the field of view (FoV) of Ritchey-Chrétien (RC) telescopes. Compared to other field correctors, the proposed solution does not affect the initial RC telescope’s aplanatic properties, and it can be added or removed from an existing RC telescope without changing the mirrors’ shape. The final system features less astigmatism, which is balanced to flatten the image surface.
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1 Introduction
The Ritchey-Chrétien (RC) is an aplanatic telescope widely used in astronomy. The RC is a variant of the Cassegrain, with the latter being a telescope that is free from spherical aberration but presents coma. This variant involves transforming the primary mirror from a paraboloid into a hyperboloid of revolution and slightly adjusting the hyperboloid secondary mirror’ shape so that the system corrects both for spherical aberration and coma [1]. However, the field curvature and astigmatism restrict the FoV of these aplanatic telescopes [2].
There are several manners to improve the RC, such as adding field correctors or using curved sensors to extend the FoV. The field lens correctors can be designed with spherical or aspheric surfaces working as a singlet or as multiple lenses [3–5]. Most of the dedicated field correctors require a modification of the telescope’s original design, for instance, by changing its parameters such as radii of curvature and conics of the mirrors [6]. The modified systems with the mirrors deviating from the hyperbolic shape are called Quasi-Ritchey-Chrétien [7]. Some lens correctors can be integrated into an existing telescope without any changes to its design [8,9]. This has an obvious advantage that such correctors can be removed when they are not needed. However, in most cases, lens correctors introduce some spherical aberration or coma when used in an unmodified telescope, making the system no longer aplanatic [10], as in the case of the Gascoigne plate [11].
The aplanatic meniscus lens proposed in this paper does not introduce spherical aberration or coma into the RC telescope for a given wavelength. This ensures that the meniscus can be inserted or removed from the system without any need to alter the telescope’s original design while preserving its aplanatic correction [12]. Furthermore, the FoV of the RC system can be significantly increased by introducing this aplanatic meniscus due to the fact that the meniscus has intrinsic astigmatism comparable to that of the telescope but of the opposite sign. As a result, total astigmatism in the RC can be reduced if the meniscus has an appropriate axial thickness and is placed in the telescope’s converging beam such that it operates in an afocal mode.
In Section 2, a method to estimate the FoV in RC telescopes is presented, and an analytical solution for the meniscus’ shape is given in Section 3. Numerical examples are given in Section 4. As shown in Section 5, the proposed meniscus can be used to extend the FoV in RC systems of various sizes, including extremely large telescopes (ELTs).
2 Astigmatism in the RC telescope
Given the radius of curvature of the primary mirror $r_1$ and the secondary magnification $m_2$, one can design a RC telescope by using Eqs. (1) to (3) to obtain the radius of curvature $r_2$ of the secondary mirror, and the conic constants $k_1$ for the primary mirror and $k_2$ for the secondary mirror [13].
where $d_1$ is the axial distance between the mirrors, $f_1$ and $f_2$ are the focal lengths of the primary and secondary mirrors, respectively, and $f$ is the focal length of the telescope. The magnification $m_2$ is given by $f/f_1$. Figure 1 illustrates the parameters used above to design a RC telescope.
Fig. 1. Optical layout of the RC telescope. The primary and secondary mirrors’ diameters are $D_1$ and $D_2$, respectively. The primary and secondary mirrors are $M_1$ and $M_2$. The distance between the primary and the secondary mirror is $d_1$. The focal length of the primary mirror and the overall system is given by $f_1$ and $f$, respectively. $F'$ is the focal point of the system. $\theta _{RC}$ is the maximum half-FoV of the RC telescope. Figure not to scale.
Using the fact that the Petzval surface is three times closer to the sagittal image surface than to the tangential surface, one can derive Eqs. (4), (6) and (7) with ray-tracing of the chief ray in the RC and applying Coddington equation for the tangential cross-section, in which, the sag of the tangential and sagittal image surfaces are $z_t$ and $z_s$, respectively, and $\theta _{RC}$ is the RC’s half FoV [14]:
withThe optimal image surface with a nearly round image spot occurs approximately midway between the tangential and sagittal surfaces, and the sag of this surface is given by:
If residual astigmatism in the RC is comparable to the system’s depth of focus, then astigmatism will have little impact on image quality [15]. The depth of focus is defined as:
where $\lambda _0$ is the primary wavelength, and $F/\#$ is the system’s f-number.In other words, equating the sag of the optimal image surface, Eq. (9), to the depth of focus, Eq. (10) and solving for tan($\theta _{RC}$), one can obtain a good approximation for the maximum half-FoV of the RC operating with a flat detector as given by:
Figure 2 demonstrates the relation between the estimated maximum half-FoV from Eq. (11), and the simulation results as function of the entrance pupil diameter for the RC. Three different $F/\#$ have been considered assuming a flat image surface for all systems. The optical designs for different $F/\#$ of the RC telescopes presented here were achieved by keeping the first mirror’s parameters constant while changing the secondary mirror.
Fig. 2. Maximum diffraction-limited half-FoV obtained on a flat detector using the Eq. (11) compared to the predicted field using simulations in OpticStudio. The entrance pupil diameter $D$ in the range of 1 - 10 m and the focal ratio $F/\#$ in the range of 8 - 12.
3 Afocal aplanatic meniscus lens
Using Fermat’s principle, one can show that a plano-convex lens can be made free from spherical aberration for a collimated beam if the lens’ convex surface is a conicoid of revolution with the eccentricity equal to the refractive index of the lens material at a given wavelength [16–19]. To collimate a converging stigmatic beam, one can use a concave-plano lens with a hyperboloidal surface, while the plano-convex lens featuring the same hyperboloidal surface will restore the original stigmatic converging beam. Furthermore, both lenses can be combined in a single meniscus lens, shown in Fig. 3, which works as an afocal lens in a converging beam. The conic constant of the hyperboloidal surfaces is given by Eq. (12):
where $n$ is the refractive index of glass at a given wavelength.Fig. 3. Ray-tracing in the aplanatic meniscus. $F'_1$ is the focal point of the RC, and $F'_2$ is the focal point of the RCm. Figure not to scale.
The lens surface height, $y$, as a function of sag $z$, is given by $y^2 = 2 Rz-(1-n^2)z^2$, where $R$ is the radius of curvature of the surface at its vertex. This equation represents a conicoid of revolution and does not require any higher-order terms of $z$ with aspheric coefficients.
We shall note that such a meniscus lens is free from spherical aberration at a given wavelength, and due to its convex-concave symmetry (the sag equation for the front and back surfaces is identical), the angular magnification is preserved for all rays in the beam before and after the lens. As a result, the lens satisfies the Abbe Sine Condition [20]. Therefore, it is also free from the aberration coma [21].
The anterior and posterior surfaces of the meniscus are identical but axially separated by non-negligible distance d so that the most likely ghost images occurring in the double path will be out of focus. Also, the two surface reflections at a level of $1 \%$ reflectivity attenuate the initial intensity in the image by a factor of $10^{-4}$.
For a given distance $z_2$ from the telescope focus $F'_1$ to the meniscus’s anterior surface, its radius of curvature $R_1$ should be chosen so that the refracted rays form a collimated beam inside the lens as shown in Fig. 3. This condition is met if the optical power of the anterior lens surface is equal to the inverse of the distance $z_2$, and from this relation, we find:
The central thickness $d$ of the meniscus and its axial position from the telescope focus determine the amount of astigmatism introduced into the RC. We shall denote an RC system combined with the aplanatic meniscus as RCm. The new focal point position $F'_2$ for the RCm system is at a distance $d$ from the initial RC focal point $F'_1$, since the radius of curvature of the posterior meniscus surface $R_2=R_1$, where $R_1$ is given by Eq. (13).
The meniscus working as an aplanatic afocal lens preserves the RC system’s aplanatic correction for a given wavelength and does not introduce any field curvature. However, astigmatism introduced by the meniscus lens’s anterior and posterior surfaces with a finite thickness does not cancel strictly, and residual astigmatism of the meniscus, as we shall see later, can be used to balance intrinsic astigmatism in the original RC system. The closer the meniscus lens is to the telescope’s focus, the more curved its surfaces become, see Eq. (13), and as a result, the less central thickness is required to produce the same amount of residual astigmatism. Given the complexity of calculating total astigmatism in the RCm system analytically, including third- and fifth-orders, it is easier to find the meniscus’s optimal central thickness that maximizes the FoV of the RCm by numerical optimization with exact ray-tracing. The meniscus corrector was designed and analyzed using the ray-tracing software OpticStudio [22].
4 Numerical example of astigmatism correction
To demonstrate the correction with an aplanatic meniscus lens, we start with an $F/10$ RC telescope with an entrance pupil diameter of $4$ $m$. The optical layout of the RCm can be seen in Fig. 4. The aplanatic meniscus corrector is introduced behind the primary mirror M1 at a distance of 2.71 m. The meniscus is located at a distance $z_1 = 11.25$ m from the vertex of the secondary mirror M2. Table 1 describes the main optical parameters of the RCm system. The axial thickness of the meniscus is $d = 13.8$ $mm$, while its optical semi-diameter is $136.0$ $mm$. The material chosen for the meniscus is fused silica since it is a high-purity glass that covers a wide spectral range of transparency between $0.2139$ $\mu m$ and $3.7067$ $\mu m$ [23]. Besides, the fused silica also exhibits high-temperature stability and low thermal shock properties, making it an optimal choice for telescopic optics [24].
Fig. 4. Optical layout of the RCm system. The primary and secondary mirrors’ diameters are $D_1$ and $D_2$, respectively. The distance between the secondary mirror and the meniscus is $z_1$. The distance between the meniscus first surface and the RC focal point $F'_1$ is $z_2$. The axial thickness of the meniscus is $d$. Figure not to scale.
Table 1. Optical and design parameters of the RCm system. The primary and secondary mirrors are M1 and M2. The anterior and posterior surfaces of the meniscus lens are R1 and R2, respectively. The image space is IMA. CO is a surface that has been used to create a central obscuration for the incoming rays in the area of the secondary mirror, its value is from 0 to 624.0 mm. The primary mirror has been set to have a circular aperture radius between 280 to 2010 mm. The entrance pupil is 4 m.
For the visible light using a wavelength range from $\lambda = 0.4861$ $\mu m$ to $\lambda = 0.6563$ $\mu m$, the refractive index of silica at the primary wavelength $\lambda = 0.6000$ $\mu m$ is $n =$ 1.458 and from Eq. (12) this corresponds to the conic constant $k= -2.126$ for both surfaces of the meniscus.
The radius of curvature of the meniscus lens is defined by the distance from the posterior surface of the meniscus to the telescope focus $z_2$ and refractive index of the lens, according to Eq. (13). Therefore, the initial position of the lens can be tested for an arbitrary distance such that say $z_1/z_2 > 10$. The thickness of the lens is used as a free parameter to correct astigmatism. However, there is a mechanical constrain on the central thickness $d$ of the lens. Here we used the manufacturing limit with $d = D/20$, where $D$ is the lens’s optical diameter. If the lens’s optimal thickness to correct astigmatism falls below this limit and becomes too thin, the lens should be placed farther away from the telescope focal point.
The main challenge in fabricating the meniscus is to keep the anterior and posterior surfaces centered while having the same radius of curvature. The fabrication method of the reflective hyperbolic concave and convex surfaces is well established, and their optical testing is well understood so that both surfaces can be tested individually in reflection. The property of the meniscus being an aberration-free lens in a converging beam offers a possibility of testing both surfaces in transmission.
Since increasing $z_2$ makes the lens less curved, this, in turn, also increases the thickness of the lens required to correct astigmatism. If the telescope operates at a broad wavelength range, the FoV might be limited by the lateral color, see discussion in Section 5. Thus, the distance $z_2$ has to be optimized by balancing astigmatism correction and controlling the lateral color. For our example $z_2 = 750$ $mm$ and $z_1/z_2 = 15.0$. The diameter of the meniscus is comparable to the linear size of the full field, and for large telescopes, it can exceed 200 mm.
The maximum full-FoV increases from $12.84$ to $17.40$ arcmins, which is $1.36$ times the FoV of the original RC system. This is equivalent to increasing the sky area by almost $1.84$ times.
The curved image surface used for the RCm is flatter than the one used for the RC system. The radius of curvature of the image surface for the RC is $-4012.6017$ $mm$, which is $2.1$ times more curved than that in the RCm system, so the lens not only corrects astigmatism but also flattens the image surface. If one uses only third-order aberrations, then Petzval curvature is not changed by the meniscus lens since $R_1 = R_2$. However, when the best image surface is considered, the field curvature is more balanced by astigmatism, and as a result, the best image surface becomes less curved.
For the numerical example, using visible light, the optical performance for the two systems is presented in Figs. 5–7. At the edge of $8.70$ arcmins half FoV, which is the maximum half FoV for a diffraction-limited RC, the RCm forms a smaller image spot. This is because the third- and fifth-order astigmatism present in the RCm are better balanced in comparison to the RC system, as shown in Fig. 6. The meniscus introduces a small positive distortion (less than 0.2 percent). We can also see in Fig. 5(b) that lateral color is present. However, it is not critical, and by narrowing the spectral band, one can reduce the image spot size even further.
Fig. 5. Spot diagrams for $(a)$ the classical RC telescope, and $(b)$ the RCm system. It shows on-axis and off-axis image spots at the maximum half FoV for the RC and RCm. The Airy disk radius is $7.32$ $\mu m$, and it is shown as a black circle.
Fig. 6. Field curvature and distortion for $(a)$ the classical RC telescope, and $(b)$ the RCm system. The $Y$-axis unit is in arcmin.
Comprehensive numerical results for this example, represented by Configuration 2 in Section 5, and two other optical designs are presented in Tables 2 and 3.
Table 2. Different configurations simulated for the RC and RCm telescopes. The entrance pupil diameter is $D_1$. FoV is the full FoV of the telescopes, and $EE80$ is the encircled energy at $80\%$. The values of the EE80 for the RC are given for the encircled energy at the maximum half-FoV of the RCm.
Table 3. Supplementary ray-tracing simulation results for the various RC and RCm optical designs. Lat. Col. is lateral color, Astig. is astigmatism, and Dist. is distortion in the system.
5 Numerical results
As mentioned above, the meniscus lens suffers from lateral color, limiting the useful FoV of the RCm, but as we show later in this section, this problem can be reduced by careful consideration of the system’s meniscus position. To keep the lateral color to a minimum, the meniscus should be placed as close as possible to the telescope focus $F'_1$. Since $R_2$ = $R_1$ in the meniscus design, the transverse lateral color in the system can be estimated using paraxial ray-tracing Eq. (14) [25]:
The chief ray angle $\bar {u}_1$ increases linearly as the lens approaches the telescope focus and becomes more curved, whereas the optimal lens thickness $d$ decreases more rapidly. Therefore, the product term $\bar {u}_1 d$ decreases. Thus, lateral color also decreases.
As the meniscus approaches the telescope focus, $z_2$ is reduced. Considering Eq. (13), for the lens to remain working in an afocal mode, the meniscus surfaces must become more curved. As a consequence, the optimal central thickness required for astigmatism correction decreases. The practical limit of how small the central thickness can be depends on the manufacturing process and structural stability of a thin glass shell. Therefore, as mentioned earlier, one has to consider a maximum permissible diameter-to-thickness ratio, which in our analysis we previously set to $D_{lens}/d = 20$.
A number of the RC telescope designs have been considered here to verify the gain in aberration-free FoV after inserting the aplanatic meniscus near the telescope focus. Figure 8 shows the relation between the maximum half-FoV obtained through ray-tracing simulations for the RC and the corrected RCm telescopes as a function of the entrance pupil diameter. Three different $F/\#$ have been considered assuming a flat image surface for all systems.
Fig. 8. The maximum diffraction-limited half-FoV attainable on a flat detector in RC and RCm systems with the entrance pupil diameter $D$ in the range of 2 - 10 m and the focal ratio $F/\#$ in the range of 8 - 12.
Scaling up the RC system will increase ray aberrations relative to the Airy disk size, which is inversely proportional to the entrance pupil diameter. As a result, the maximum FoV for a diffraction-limited system is reduced in a non-linear manner, see Eq. (11). The most prevalent aberration in an RC system is astigmatism, and after being corrected with the help of an aplanatic meniscus, the remaining field curvature limits the field. Another limiting factor is the lateral color introduced by the meniscus. The image spot size in an RCm system is affected by lateral color, but it can be kept smaller than the original image spot size imposed by intrinsic astigmatism in the RC system.
We should note that the FoV gain is somewhat reduced with the increase of the entrance pupil diameter. This happens because as the system’s size increases, the diameter of the meniscus increases, and due to the diameter-to-thickness ratio limitations, the meniscus has to be placed farther away from the focal point of the RC so that the astigmatism is not over-corrected. Therefore, the lateral color increases, becoming the main limiting factor of the system. Also, it is important to mention that the increase in FoV is still significant with the use of the aplanatic meniscus in those cases.
A trend can be noticed in Fig. 8. Faster systems produce higher FoV gains when the meniscus is used. This occurs because the ratio of the diameter of the meniscus to its thickness, $D_{lens}/d = 20$, is limited to 20. When comparing telescopes with the same diameter $D$ but differing $F/\#$, the RC astigmatism increases noticeably. As a result, a thicker meniscus is required, which allows the lens to be positioned closer to the focal point while keeping the $D_{lens}/d = 20$ ratio, decreasing lateral color.
To find the optimal minimum thickness of the meniscus, $d_0$, depending on the position of the meniscus, a range of entrance pupil diameters for an $F/10$ telescope have been simulated using a flat image plane for different values of Z, which is the ratio between the distance of the meniscus to the secondary mirror and to the focal point $F^\prime _1$, as defined in Eq. (15). Our findings can be seen in Fig. 9. Thus, the closer the meniscus approaches the focal point $F_1^\prime$, the thinner the meniscus must be to balance astigmatism in the RCm system.
Fig. 9. The minimum thickness for the meniscus to correct astigmatism for different $F/10$ telescope entrance pupil diameters using a flat image surface as a function of the ratio $Z$ = $z_1$/$z_2$, see Fig. 4. The diameter-to-thickness ratio is kept at 20.
It is important to note that these results take into the constrain in terms of diameter-to-thickness ratio, which is kept smaller than 20 for the lens manufacturability.
When diameter-to-thickness criteria are not used, the FoV starts to decline for higher Z values. As mentioned before, the thickness of the meniscus needed to correct astigmatism diminishes due to the lens surfaces being more curved. Otherwise, the overall astigmatism is over-corrected. Astigmatism compensation requires two optimal factors, the radius of curvature of the surfaces of the meniscus and their axial separation $d$.
The closer the meniscus moves towards the primary mirror, i.e., farther away from the focal point, the less curved the surfaces will be. Consequently, its central thickness must be increased to correct the RC’s astigmatism, thus increasing the lateral color accordingly. For greater Z values, the meniscus is closer to the focal point. Hence, the surfaces are more curved, requiring less central thickness to balance astigmatism, and increasing the diameter-thickness ratio. If the diameter-to-thickness ratio complies with the manufacturing limits, then at some point, the meniscus thickness might become greater than the optimal thickness $d_0$, and astigmatism will be over-corrected.
Overall astigmatism in the system, up to the seventh-order, has been calculated using the Zernike coefficients as stated in 17. The Noll notation was chosen for the Zernike polynomial, where the first index is 1, and the cosine and sine terms are even and odd, respectively [26].
Configurations 1 to 3 correspond to the telescopes with an $F/\#$ = 10 and the system entrance pupil diameter equal to 2, 6, and 8 m, respectively. The results from Tables 2 and 3 demonstrate a significant improvement in FoV size of the original RC designs, showing an increase of up to $1.79$ times in the object area from an RC telescope and a major reduction in total astigmatism. Furthermore, in every optical design reviewed, meniscus distortion is below $0.3\%$. Thus, image scale is not compromised.
The effect of the aplanatic meniscus in the exit pupil image quality is smaller than $1\%$. Therefore it is minimal in terms of monochromatic aberrations. Furthermore, as the meniscus has no optical power, the chromatic effect in the exit pupil image quality is also minimal.
As can be seen in Tables 2 and 3, the maximum FoV demonstrates a larger FoV gain for a flat image surface. We should note that the most effective configuration for FoV gain is the first one, in a system with a $2m$ entrance pupil diameter and a flat image plane. This happens because the curved image surface can correct the field curvature in both systems. However, as the final image in the RCm is already flatter than the one in the RC, the curved image surface has less impact on the RCm. Consequently, the FoV increase from the RC, and the RCm is smaller. Nevertheless, a less curved sensor can be desirable for manufacturing purposes.
For the telescope with a typical diameter D=4 m, the decentration and tilt should not exceed 3.5 mm and 0.5 degrees, respectively. Otherwise, the image will be affected especially by defocus, tilt, and astigmatism. And the axial shift of the meniscus should not exceed 10 mm to avoid introducing spherical aberration, coma, and astigmatism.
6 Numerical optimization alternative
We shall optimize the meniscus parameters (central thickness and the radius of curvature of the surface) with exact ray-tracing to further increase the FoV of the RCm. This allows us to obtain additional gain in the FoV, which can be doubled. However, this approach will lead to a deviation of the aplanatic properties of the system. Figure 10 shows the results of the numerical example from Section 4, the entrance pupil diameter is 4 m, the meniscus is re-optimized for $d$ and $R_1$. The optimized parameters obtained for the RCm are demonstrated in Table 4. It is worth noticing that the radius of curvature and the conic constant of both surfaces were kept equal for optimisation stability.
Fig. 10. Simulation results for the modified RCm, showing $(a)$ the encircled energy at 80 $\%$, $(b)$ the spot diagram, and $(c)$ the field curvature.
Table 4. Parameters used for the modified RCm. The parameters numerically optimized are marked in bold.
In this example, the maximum full-FoV increases from $13.2$ arcmins of the original RC to $25.2$ arcmins in the RCm optimized, representing a $1.91$ times increase in FoV and $3.64$ times in object area. A noticeable coma can be seen in spot diagram Fig. 10(b) for the intermediate field point. Thus, as expected, the system is no longer aplanatic. However, the FoV with diffraction-limited image quality is significantly increased.
7 Conclusion
A meniscus has been analytically designed to increase the FoV of RC telescopes. This approach does not disrupt their aplanatic imaging solution. By increasing the FoV, the object area size can be significantly increased by a factor of 1.5 to 4 times, depending on the entrance pupil size and its image surface shape. The advantage of the aplanatic meniscus solution over other approaches is in its convenient integration and removal on existing RC telescopes without the need of remodeling their optical design. Given manufacturing limits for large and thin meniscus lenses, the meniscus’ optimal position to minimize lateral color and achieve a larger FoV has also been considered. Besides, an equation estimating the approximated maximum FoV of an RC telescope has also been presented.
The RCm can be designed with different entrance pupil sizes and $F/\#$, showing promising results for telescopes with an entrance pupil up to $8$ $m$. However, because of the smaller FoV of larger systems and the lateral color added with the thicker meniscus, the telescopes with a larger entrance pupil display a lower gain in FoV compared to the smaller than $8$ $m$ class telescopes.
Funding
FotoNation and Science Foundation Ireland (ID: 13/SPP/I2868).
Acknowledgements
The authors would like to thank Prof. Christopher Dainty for his support of this research.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. P. Bely, The design and construction of large optical telescopes (Springer, 2003).
2. M. Bahrami and A. V. Goncharov, “All-spherical catadioptric telescope design for wide-field imaging,” Appl. Opt. 49(30), 5705–5712 (2010). [CrossRef]
3. R. N. Wilson, “Corrector systems for cassegrain telescopes,” Appl. Opt. 7(2), 253–263 (1968). [CrossRef]
4. C. G. Wynne, “Field correctors for large telescopes,” Appl. Opt. 4(9), 1185–1192 (1965). [CrossRef]
5. S. Rosin and M. Amon, “Extending the stellar field of view of ritchey-chretien telescopes,” Appl. Opt. 11(7), 1623–1629 (1972). [CrossRef]
6. C. G. Wynne, “Iv field correctors for astronomical telescopes,” Prog. Opt. 10, 137–164 (1972). [CrossRef]
7. V. Terebizh, Survey Telescope Optics (SPIE, 2019).
8. M. Bahrami and A. V. Goncharov, “The achromatic design of an atmospheric dispersion corrector for extremely large telescopes,” Opt. Express 19(18), 17099–17113 (2011). [CrossRef]
9. M. C. Rocha and A. V. Goncharov, “A quasi-aplanatic refractive pair for correcting astigmatism in ritchey-chrétien telescopes,” Proc. SPIE 11871, 1187108 (2021). [CrossRef]
10. R. N. Wilson, Reflecting Telescope Optics I: Basic Design Theory and its Historical Development (Springer, 2013).
11. S. C. B. Gascoigne, “On ritchey-chretien systems,” The Observatory 85, 79–81 (1965).
12. M. C. Rocha and A. V. Goncharov, “An aplanatic meniscus lens for correcting astigmatism in ritchey-chrétien telescopes,” Proc. SPIE 11451, 285 (2020). [CrossRef]
13. H. Gross, F. Blechinger, and B. Achtner, Handbook of optical systems, volume 4: Survey of Optical Instruments (Wiley, 2008).
14. J. Braat and P. Török, Imaging optics (Cambridge University, 2019).
15. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).
16. G. Kweon and C. Kim, “Aspherical lens design by using a numerical analysis,” J. Korean Phys. Soc. 51(1), 93–103 (2007). [CrossRef]
17. R. G. González-Acuña and H. A. Chaparro-Romo, “General formula for bi-aspheric singlet lens design free of spherical aberration,” Appl. Opt. 57(31), 9341–9345 (2018). [CrossRef]
18. R. G. González-Acuña, H. A. Chaparro-Romo, and J. C. Gutiérrez-Vega, “Exact equations for stigmatic singlet design meeting the abbe sine condition,” Opt. Commun. 479, 126415 (2021). [CrossRef]
19. A. Silva-Lora and R. Torres, “Superconical aplanatic ovoid singlet lenses,” J. Opt. Soc. Am. A 37(7), 1155–1165 (2020). [CrossRef]
20. A. Ernst, “On the estimation of aperture in the microscope,” J. R. Microsc. Soc. 1(3), 388–423 (1881). [CrossRef]
21. R. Kingslake and R. B. Johnson, Lens design fundamentals (Academic, 2009).
22. “Zemax opticstudio premium,” Zemax LLC, 21.3.1 (2021).
23. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. A 55(10), 1205–1209 (1965). [CrossRef]
24. N. P. Bansal and R. H. Doremus, Handbook of glass properties (Elsevier, 2013).
25. V. N. Mahajan, Optical imaging and aberrations: Ray geometrical optics, vol. 45 (SPIE, 1998).
26. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66(3), 207–211 (1976). [CrossRef]
