1 Introduction

The solar corona is probably one of the most enigmatic regions of our star. Many of the physical processes occurring in this layer are not fully understood yet (e.g., coronal heating or the formation of coronal mass ejections). To get a better insight of the mechanisms that drive these phenomena, observations of the corona through specialized astronomical instruments (coronagraphs) are needed. Observing the corona poses a significant challenge as the solar disk is about \(10^{6}-10^{10}\) times brighter [1]. Coronagraphs must therefore make use of one or several occulters to block light coming directly from the disk [2, 3]. Yet, diffraction produced by the occulter spreads solar disk light all over the field of view. Even a small fraction of spread light can produce a halo bright enough to falsify coronal observations [4]. Usually, requirements for the fraction of acceptable stray (unwanted) light from the disk to the corona are as strict as \(10^{-8}\) [5]. A reduction of stray light to such low levels is achieved by means of masks, baffles and Lyot stops positioned at strategic locations along the optical path of the instrument.

Apart from diffraction, there are other sources of stray light that need to be addressed. Scattering produced by the surface roughness of the different optical elements and ghost images caused by multiple reflections between them also spread light from the disk to the corona and can produce artificial signals with amplitudes as high as \(10\,\%\) of the coronal brightness [6]. Several strategies have been devised to minimize the amount of scattered stray light and ghosts. Some of them include the use of super polished optical elements with surface microroughness below 1 nm root mean square (RMS) to reduce scattering [7] or avoiding air gaps between lenses to reduce the number of reflections that can potentially create ghosts [5]. Keeping the design simple is therefore key to achieving the best possible performance. The fewer the number of optical elements within the instrument, the smaller the contribution of scattering and ghost images on the final image.

Narrowband filters centered on a spectral line of interest are one of the optical elements often employed in coronagraphs. Filters are sometimes placed at one of the images of the telescope pupil while illuminated with collimated light (e.g., the collimated configuration described in [8]). This configuration allows both to average out local defects (e.g., dust) and to achieve a homogeneous optical quality over the whole field of view (FOV). Collimated setups suffer from a variation of the incidence angle on the filter that shifts its bandpass over the field of view [9]. Long focal lengths of the collimator can reduce the incident angle and therefore the impact of this blueshift, at the expense of increasing the length of the system. Telephoto lenses can therefore be employed with the aim of keeping both the instrument dimensions and the incident angles on the filter under control.

Spaceborne instruments can greatly benefit from using telephoto lenses due to the limited space available for their allocation on the spacecraft. This is the case of the Coronal Magnetograph (CMAG) mission [10], whose conceptual design includes a Fabry-Pérot filter. CMAG is a spaceborne mission proposed to ESA in 2022 as a Fast mission. The mission is designed to perform spectropolarimetric measurements from the Lagrangian L5 point using a coronal magnetograph together with an external occulter. In CMAG, a collimator focuses an intermediate telecentric image of the solar corona to infinity and projects the telescope pupil at its back focal plane, where the filter is placed. A second telecentric camera lens reimages the solar scene on the instrument detector with the desired plate scale. Both the collimator and the camera lenses have very long focals (\(\sim\) 1 m) and must be designed as telephoto lenses to comply with the volume requirements of the mission. However, conventional telephoto lenses shorten the length from the first lens to the back focal plane, at the cost of increasing its front focal length. The collimator and camera lenses must be designed in this instrument to reduce their front-to-back focal distances, instead of just the back focal length. Meanwhile, the design philosophy of CMAG is to keep the number of lenses and air gaps between them to the minimum with the goal of reducing scattered stray light and ghost images at the detector, as well as to optimize the throughput of the instrument.

In this paper we explore the design principles and performance of two-component telephoto lenses for which the front-to-back focal plane distance is minimized. Our main goal is to find a good compromise between the length of the telephoto lens and the number of lenses that are employed. Our paper is structured as follows. In Section 2, we introduce the paraxial equations of telephoto lenses that reduce their front-to-back focal lengths. In Section 3, we propose a methodology to design a telephoto lens based on two cemented doublets for the CMAG instrument. We assess the optical quality of the instrument by identifying and analyzing the main aberrations. In Section 4, we athermalize the system for an operational temperature range between \(-20\)°C and \(+20\)°C. In Section 5, we apply a tolerance analysis to study the feasibility of our proposal. Finally, Section 6 summarizes the main results.

2 Front-to-back focal telephoto equations

Telephoto lenses are usually made of two separated components. The focal lengths of each component, \(f'_a\) and \(f'_b\), can be expressed as

$$\begin{aligned} f'_a = \frac{DF}{F-B}, \end{aligned}$$
(1)
$$\begin{aligned} f'_b = \frac{-DB}{F-B-D}, \end{aligned}$$
(2)

where D is the separation between components, F is the effective focal length of the system and B is its back focal length [11].

Figure 1 illustrates the layout of a conventional telephoto lens consisting of a converging lens followed by a diverging lens.

Fig. 1
figure 1

Conventional telephoto lens diagram. L is the total length, F is the effective focal length, D is the separation between components, B is the back focal length and A is the front focal length. Dimensions not to scale

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The total length of a telephoto lens, L, can be expressed as a fraction of the effective focal length in the way

$$\begin{aligned} L = KF, \end{aligned}$$
(3)

where K is the so-called telephoto ratio [12]. Whenever K is below unity, the distance from the first lens to the back focal plane is reduced with respect to the effective focal length, but its front focal length, A, increases. The entrance pupil of image-space telecentric optical systems is located at the front focal plane, where our filter is placed. Hence, we must consider the total length of the system as its front-to-back focal length. The distance from the front to the back focal planes (neglecting lens thicknesses), \(\tilde{L}\), is simply given by

$$\begin{aligned} \tilde{L} = A + D + B. \end{aligned}$$
(4)

Figure 2 shows \(\tilde{L}\) as a function of the telephoto ratio between 0 and 1, and for different values of the separation between components, in units of F. The front-to-back focal distance increases as K decreases, and is always greater than 2F when \(K<1\). Note that 2F represents the total length of the system in the single-lens (\(K=1\)) case. Therefore, in our case, the total length of the collimator is actually increased when using a conventional telephoto lens.

Fig. 2
figure 2

Front-to-back focal distance as a function of K for different values of D/F between 0 and 1

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To reduce \(\tilde{L}\) below 2F, we can redefine the telephoto ratio to take into account A as

$$\begin{aligned} \tilde{K}F = A + D + B, \end{aligned}$$
(5)

where \(\tilde{K}\equiv \tilde{L}/F\) and the front focal length is given by [11]

$$\begin{aligned} A = \frac{F(f'_b-D)}{f'_b}. \end{aligned}$$
(6)

Using (2), this expression can be rewritten as

$$\begin{aligned} A = \frac{F(F-D)}{B}. \end{aligned}$$
(7)

From (5), (7), we obtain the following quadratic equation

$$\begin{aligned} B^2 + B(D-\tilde{K}F) + F(F-D) = 0. \end{aligned}$$
(8)

whose solution is

$$\begin{aligned} B = \frac{(\tilde{K}F-D)\pm \sqrt{(D-\tilde{K}F)^2-4F(F-D)}}{2}. \end{aligned}$$
(9)

Substituting (9) into (1), (2), we can express the focal lengths of each component as a function of \(\tilde{K}\), F and D as

$$\begin{aligned} f'_a = \frac{2DF}{F(2-\tilde{K})+D\pm \sqrt{(D-\tilde{K}F)^2-4F(F-D)}}, \end{aligned}$$
(10)
$$\begin{aligned} f'_b = \frac{D(D-\tilde{K}F)\pm D\sqrt{(D-\tilde{K}F)^2-4F(F-D)}}{F(2-\tilde{K})-D\pm \sqrt{(D-\tilde{K}F)^2-4F(F-D)}}. \end{aligned}$$
(11)

Our re-defined telephoto ratio, \(\tilde{K}\), must be greater than unity to allow the separation between the two lenses to be smaller than the total length of the system (\(D<\tilde{L}\)). Meanwhile, to reduce the length of the system compared to a single-component image-space telecentric lens, the distance from the entrance pupil to the image plane must be below 2F. Therefore, the value of \(\tilde{K}\) must be in the range

$$\begin{aligned} 1<\tilde{K}<2. \end{aligned}$$
(12)
Fig. 3
figure 3

Focal lengths of each component of the telephoto lens as a function of D for different values of \(\tilde{K}\) between 1 and 2, calculated using the positive (a) and negative (b) square root of (10), (11)

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Figure 3 shows \(f'_a\) and \(f'_b\) as a function of the separation between each component of the telephoto lens, for different values of \(\tilde{K}\) between 1.1 and 1.9 in steps of 0.2. The focal lengths of Fig. 3a were calculated using the positive solution of the square root of (10), (11), while Fig. 3b corresponds to the negative solution. The focal lengths, \(f'_a\) and \(f'_b\), and the separation between components, D, have been normalized with respect to the focal length of the system, F. The vertical dashed lines indicate the minimum value of D/F for which (10), (11) are defined. Below this limit, the square root has no real solutions. This minimum separation between components decreases as the telephoto ratio increases. Note also that:

  1. 1.

    The results obtained with both signs of the square root are equivalent in the paraxial approximation, where the choice between the positive and negative solution only affects the order of the telephoto lens components. Note that in a real design, the beam footprint is larger on the component closer to the pupil, making it more sensitive to certain aberrations (e.g., coma and spherical aberration).

  2. 2.

    \(f'_a\) and \(f'_b\) are positive for values of \(\tilde{K}\) between 1 and 2, while in a conventional telephoto lens the first component is converging and the second is diverging.

  3. 3.

    For the positive solution of the square root, \(f'_a\) decreases with D, while \(f'_b\) increases with D. The opposite occurs if we choose the negative solution of the equation.

  4. 4.

    The focal lengths of both components increase with the telephoto ratio.

3 Real case application: CMAG collimator

The collimator of CMAG focuses an image of the corona located at the telecentric focal plane of the telescope into infinity, while projecting the telescope pupil at its back focal plane, where an etalon is placed. We have designed this collimator as an imaging lens, with the telescope pupil located at its front focal plane and the telecentric image of the corona formed at its back focal plane. Table 1 summarizes its design requirements.

Table 1 Instrument specifications
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The CMAG spacecraft will have active thermal control [10]. The entrance windows of the instrument will significantly reduce the heat load from solar radiation, transmitting only a narrow wavelength range centered around the target spectral lines. Some optical components of CMAG, such as the narrowband interference filters on the filter wheel, the liquid crystal variable retarders used for the polarization modulation and the etalon, are extremely sensitive to temperature. These elements require high thermal stability (\(\sim \pm 0.05\)°C for the etalon) at their operational temperature. However, the collimator is less sensitive to temperature variations and therefore does not require a dedicated thermal control system. For the collimator design, it is assumed that the spacecraft’s active thermal control will maintain an operational temperature range of [−20°C, \(+\)20°C].

The collimator must also be designed as a two-lens system to minimize the sources of unwanted stray light and ghosts while maximizing the throughput of the instrument. It must be also telephoto to reduce the distance from the image plane to the etalon.

We have employed \(\tilde{K} = 1.8\) and a separation between components \(D = 0.8F\) as the starting point for the design. This choice of \(\tilde{K}\) and D gives long focals for the two lenses, which reduce the incidence angles on the lenses surfaces. Using (10), (11), we get that

$$\begin{aligned} f'_a = 995\;\textrm{mm}, \end{aligned}$$
(13)
$$\begin{aligned} f'_b = 2605\;\textrm{mm}, \end{aligned}$$
(14)

where the positive solution of the square root of (10), (11) has been arbitrarily chosen. Selecting the negative solution would reverse the orientation of the system, interchanging the back focal length with the front focal length.

As starting point, we have decided to use cemented achromatic doublets to reduce the number of lenses and air gaps of the system. We have selected N-SK16 and N-SF4 as crown and flint glasses, respectively. This pair of glasses is adequate to obtain achromatized doublets due to the significant difference in their Abbe numbers.

The focal lengths of each lens for an achromatic doublet, \(f'_1\) and \(f'_2\), are given by

$$\begin{aligned} f'_1 = \frac{\nu _1-\nu _2}{\nu _1}f', \end{aligned}$$
(15)
$$\begin{aligned} f'_2 = \frac{\nu _2-\nu _1}{\nu _2}f', \end{aligned}$$
(16)

where \(f'\) is the focal length of the doublet and \(\nu _1\) and \(\nu _2\) are the Abbe numbers of the first and second singlet, respectively [13]. The radii of each surface of the singlets, \(R_1\) and \(R_2\), can be calculated from

$$\begin{aligned} \frac{1}{f'_s} = (n-1)\left( \frac{1}{R_1}-\frac{1}{R_2}\right) +\frac{(n-1)^2}{n}\cdot \frac{d}{R_1R_2}, \end{aligned}$$
(17)

where \(f'_s\) is the focal length of each singlet, n is the refractive index of the glass and d is the center thickness [13]. Table 2 shows the radii of each surface of the doublets which are necessary to obtain the focal lengths, assuming that the first lens of both doublets is biconvex (i.e., \(R_1 = -R_2\)) and that the singlets are cemented (each doublet consists of three surfaces).

Table 2 Radii of each surface of the cemented doublets
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The optical system has been simulated in Zemax OpticStudio [14] using the radii shown in Table 2. The initial center thickness of all singlets has been set to 5 mm and the separation between each doublet, D, is \(720\;\textrm{mm}\). The aperture stop is located at the front focal plane to guarantee that the system is image-space telecentric, and its diameter, \(D_{\textrm{PE}}\), is \(45\;\textrm{mm}\), so that the f-number of the instrument is 20. The image quality of the optical system has been evaluated at three fields: 0° (on-axis), 0.896° (zonal field) and 1.28° (maximum field). The effective focal length, F, of the optical system obtained with these parameters is \(902\;\textrm{mm}\), and its total length, L, is \(1630\;\textrm{mm}\).

The system has been optimized to obtain the same effective focal length and total length of the paraxial design while minimizing the RMS of the spot diagram, setting the radii and the thickness of each surface as free parameters. We have imposed image-space telecentricity and that the focal lengths of each component of the telephoto lens are equal to (13), (14).

Fig. 4
figure 4

Spot diagram of the preliminary real design

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Figure 4 shows the spot diagram on the image plane. Note that the size of the spot is contained within the Airy disk only for the on-axis field. The RMS wavefront errors, \(\Delta W\), for each field are given in Table 3. The value of \(\Delta W\) is greater than \(\lambda /20\) for the zonal and maximum fields.

Table 3 RMS wavefront error for each field (\(\tilde{K}=1.8\))
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3.1 Optimization

Fig. 5
figure 5

Flowchart of the sequential optimization of the telephoto lens

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We have optimized the system to minimize its total length while ensuring that the image quality requirement (\(\Delta W < \lambda /20\)) is satisfied. We have then reduced the telephoto ratio sequentially from 1.8 to 1.0 in steps of 0.1, with the aim of studying how the performance of the system varies with \(\tilde{K}\), and determining the lowest acceptable telephoto ratio. Figure 5 shows a flowchart of the optimization process we followed. Starting from the optimized system for \(\tilde{K}=1.8\), we reduce the value of \(\tilde{K}\) by 0.1 in the merit function and optimize again. We have repeated this process until \(\tilde{K}=1.0\).

Since the Fabry-Pérot interferometer is located at the position of the entrance pupil, the distance from the entrance pupil to the first surface of the optical system, A, must be sufficiently large to allow the positioning of the filter. Figure 6 shows A as a function of the telephoto ratio. The value of A decreases almost linearly when \(\tilde{K}\) is reduced down to \(\tilde{K}\sim 1.3\). Below this value, A is virtually zero. We therefore added operands to the merit function to control A, and to limit the thickness of each lens. In this stage of the optimization, the constraints on the focal lengths of each component of the telephoto lens were removed.

Fig. 6
figure 6

Distance from the entrance pupil to the first surface of the optical system as a function of telephoto ratio

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The wavefront error (WFE) was initially used as the image quality criterion for the merit function, as it is an adequate metric to evaluate diffraction limited optical systems. However, we observed that the WFE improved more when optimizing the spot size if the telephoto ratio was reduced. We therefore opted to minimize the spot size instead for the sequential optimizations.

3.2 Analysis of the optical performance

Figure 7 shows the dependence of the geometric spot radius and the RMS wavefront error with \(\tilde{K}\). The geometric spot radius is smaller than the Airy disk radius for all values of \(\tilde{K}\), but the RMS wavefront error is greater than \(\lambda /20\) for the on-axis field for \(\tilde{K}=1.0\). Note that image quality significantly drops when \(\tilde{K}\) is lower than 1.1, which suggests that the system becomes unstable for telephoto ratios approaching unity. We think that the best compromise between compactness of the system and optical performance is found for \(\tilde{K}=1.1\). This telephoto ratio achieves a \(45\,\%\) reduction in the total length of the optical system when compared to the case in which the collimator is not designed as a telephoto system.

Fig. 7
figure 7

Geometric spot radius (top) and RMS wavefront error (bottom) as a function of telephoto ratio

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Figure 8 shows the full layout of the optical design after its optimization. Table 4 shows the RMS wavefront error for \(\tilde{K}=1.1\) after a final optimization using \(\Delta W\) as the image quality criterion for the merit function. This last optimization slightly improves the image quality.

Fig. 8
figure 8

Layout of the optimized optical design. The telephoto ratio is 1.1

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Table 4 RMS wavefront error for each field after optimizing the optical system (\(\tilde{K}=1.1\))
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Figure 9 shows the primary (third-order) aberrations of the final optical system. The major aberration is distortion, followed by field curvature (i.e., Petzval curvature), astigmatism and spherical aberration. Yet, distortion is smaller than \(0.2\,\%\). This low distortion is characteristic of telecentric systems. Note also that the amount of coma is negligible.

Fig. 9
figure 9

Seidel diagram of the third-order aberrations introduced by each surface and their total sum. Maximum aberration scale is \(3\;\mathrm {\upmu m}\). Grid lines are spaced \(0.3\;\mathrm {\upmu m}\)

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Fig. 10
figure 10

Variation of the spot diagram with the position of the image plane for the optimized telephoto lens. The zero position represents the best focus found after optimizing the system

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Figure 10 shows the spot diagram across the FOV, as well as the corresponding Airy disk. The spot radius is contained within the Airy disk (diffraction-limited) for all fields. It is worth noting that the best image for the on-axis field occurs when the image plane is slightly separated from its original position due to the field curvature. Astigmatism plays a significant role at the maximum field: the tangential focus is at the position of the image plane, while the sagittal focus is approximately 150 \(\mathrm {\upmu m}\) to the left of the image plane.

4 Athermalization

The instrument must operate properly between \(-20\)°C and \(+20\)°C. Consequently, the system must be athermalized so that the image quality requirement is fulfilled throughout the operational temperature range. The thermal expansion coefficients, \(\alpha\), for each component of a cemented doublet must be similar in order to reduce mechanical stress in the glasses caused by temperature changes. We have set the following requirement for \(\alpha\) values:

$$\begin{aligned} \left| {\alpha _1 - \alpha _2}\right| < 10^{-6}/\textrm{K}, \end{aligned}$$
(18)

where \(\alpha _1\) and \(\alpha _2\) are the thermal expansion coefficients of each component of a cemented doublet. The initial glasses (N-SK16 and N-SF4) do not satisfy this requirement \(\left( \left| {\alpha _1 - \alpha _2}\right| = 3.2\cdot 10^{-6}/\textrm{K}\right)\). We have therefore selected a catalog of commonly used in space applications glasses with similar thermal coefficients and found the best pair by allowing glasses to be substituted during the optimization. Table 5 shows the selected catalog.

Table 5 Glass catalog used for the athermalization of the optical system
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The system has been optimized for three temperature configurations, namely 20°, 0° and \(-20\)°, with the condition that its effective focal length remains constant for all configurations (\(F=900\;\textrm{mm}\)). The optimal choice of glasses has been found to be N-BK7 and S-BSM18 for both doublets. The variation of the effective focal length throughout the operational temperature range after the athermalization of the system is \(5\;\mathrm {\upmu m}\). Figure 11 shows the spot diagram for each field. The spot size is comparable to that shown in Fig. 10, and diffraction-limited performance is still achieved.

Fig. 11
figure 11

Spot diagram of the athermalized telephoto lens

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Figure 12 shows the variation of the RMS wavefront error with temperature, T, in the range \(-20\)°C to \(+20\)°C. Image quality is virtually constant across the entire temperature range for all fields. The RMS wavefront error remains below \(\lambda /20\), and refocusing is not required to maintain diffraction-limited performance.

Fig. 12
figure 12

RMS wavefront error as a function of temperature

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5 Tolerance analysis

Manufacturing tolerances determine the maximum allowed errors during the fabrication and assembly of the components of an optical system. Extremely strict tolerances can increase significantly the cost of the instrument. On the other side, the image quality requirement might not be satisfied if tolerances are exceedingly loose.

We have performed a preliminary tolerance analysis using the “easy” (standard) tolerance values defined in [15], with the image plane as a compensator. We have found that the RMS wavefront error is greater than \(\lambda /20\) for this choice of tolerances. The increase of the WFE is primarily caused by the irregularities on the surfaces of the first doublet. Hence, we reduced their tolerances from 1 fringe to 0.4 fringes to limit the value of \(\Delta W\). Table 6 shows the final tolerances of each element.

Table 6 Final tolerances of the telephoto lens
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Fig. 13
figure 13

Histogram of the RMS wavefront error obtained in the Monte Carlo analysis for 1089 cycles

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Table 7 RMS wavefront error and compensator statistics of the Monte Carlo analysis
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After establishing the tolerances based on the results of the sensitivity analysis, we performed a Monte Carlo analysis using 1089 cycles (i.e., the square of the number of operands used to set the tolerances). Figure 13 shows the histogram of the results. Note that the maximum RMS wavefront error is always below \(\lambda /20\) (\(0.05\lambda\)). Table 7 summarizes the statistics of the Monte Carlo analysis. The mean RMS wavefront error is approximately \(\lambda /50\) and the best result corresponds to a \(\Delta W\) of \(\lambda /100\). The absolute value of the variation of the image plane position, \(\Delta B\), is always smaller than 12 mm.

6 Summary and conclusions

We have derived the equations that describe two-component telephoto lenses considering the distance from the front to the back focal planes. We have redefined the telephoto ratio, \(\tilde{K}\), to take into account the front-to-back focal distance. Our telephoto ratio is defined in the range \(1<\tilde{K}<2\). Values above \(\tilde{K}=2\) do not decrease front-to-back focal length. Values of \(\tilde{K}\) below unity can be achieved only if a third lens is added. A third lens introduces an additional degree of freedom, which can be used to project a virtual pupil on its focal plane. This way the telephoto ratio can be reduced below unity. In this work, we avoid using three components to prevent additional reflections, which could increase stray light, introduce ghost images and reduce the total throughput.

We have applied these equations to design a collimator for CMAG, a coronal magnetograph used to infer magnetic fields in the inner corona from the spectropolarimetric analysis of emission lines. We have selected this instrument because a filter is placed at the back focal plane of a collimator with a long focal length as large as 900 mm. We have designed the collimator as an image-space telecentric telephoto lens. We have reduced the value of \(\tilde{K}\) gradually from 1.8 to 1.0 to minimize the distance between the entrance pupil and the image plane. We found that image quality decreases abruptly when \(\tilde{K}\) is below 1.1. Based on the RMS wavefront errors obtained for each telephoto ratio, we have selected \(\tilde{K}=1.1\) for our final system. By employing this telephoto ratio, the total length of the system is reduced by \(45\,\%\).

We have analyzed the optical performance of the optimized system. The instrument is diffraction limited for all fields (on-axis, zonal and maximum) and the RMS wavefront error is smaller than \(\lambda /20\). The main aberrations are distortion, field curvature, astigmatism and spherical aberration. Distortion is smaller than 0.2 %, despite being the most predominant aberration.

We have athermalized the system for temperatures ranging from \(-20\)°C to \(+20\)°C, selecting glasses which are typically used in spaceborne instrumentation. After the athermalization, the system is still diffraction limited, and the effective focal length is virtually constant throughout the operational temperature range.

We have finally studied the impact of fabrication tolerances. We have determined the appropriate values of manufacturing tolerances based on the results of a sensitivity analysis. We have found that the most critical tolerances are the irregularities on the surfaces of the first doublet. In addition, a Monte Carlo analysis confirmed that, using standard tolerances —except for the surface roughness of the first doublet—, the RMS wavefront error is below \(\lambda /20\) and thus the image quality requirement is satisfied.