## Abstract

This paper is devoted to the improvement of ground-based telescopes based on diffractive primary lenses, which provide larger aperture and relaxed surface tolerance compared to non-diffractive telescopes. We performed two different studies devised to thoroughly characterize and improve the performance of ground-based diffractive telescopes. On the one hand, we experimentally validated the suitability of the stitching error theory, useful to characterize the error performance of subaperture diffractive telescopes. On the other hand, we proposed a novel ground-based telescope incorporated in a Cassegrain architecture, leading to a telescope with enhanced performance. To test the stitching error theory, a 300 mm diameter, 2000 mm focal length transmissive stitching diffractive telescope, based on a three-belt subaperture primary lens, was designed and implemented. The telescope achieves a 78 cy/mm resolution within 0.15 degree field of view while the working wavelength ranges from 582.8 nm to 682.8 nm without any stitching error. However, the long optical track (35.49 m) introduces air turbulence that reduces the final images contrast in the ground-based test. To enhance this result, a same diameter compacted Cassegrain ground-based diffractive (CGD) telescope with the total track distance of 1.267 m, was implemented within the same wavelength. The ground-based CGD telescope provides higher resolution and better contrast than the transmissive configuration. Star and resolution tests were experimentally performed to compare the CGD and the transmissive configurations, providing the suitability of the proposed ground-based CGD telescope.

© 2017 Optical Society of America

## 1. Introduction

The interest of using diffractive telescopes for astronomical observation has been widely discussed in recent years. For instance, by using a membrane based diffractive optical element (DOE) to serve as the primary lens in space telescopes, not only the launching weight is dramatically reduced, but also a loose surface shape tolerance is achieved [1–4]. The concept of the diffractive telescope was firstly proposed by D. Buralli within a Keplerian configuration [5]. Afterwards, further studies on diffractive telescopes were proposed by introducing Schupmann chromatic dispersion correction theory [6], as provided by R. Hyde [7], with an optical arrangement including two separate parts: (*i*) a large diameter diffractive lens used as the telescope aperture; and (*ii*) a chromatic dispersion correction optical system serves as the eyepiece.

Nowadays, ground-based diffractive telescopes are extensively investigated for their relations to future astronomical applications, such as X-ray diffraction astronomy observation [8], or compact membrane space satellites [9]. Under this scenario, L. Koechlin et al. further investigated the diffractive telescopes [10,11], provided the “generation II” ground-based prototype, which includes a 20 cm aperture, 18 m focal length Fresnel lens with a working wavelength of 800 nm. Following, P. Atcheson et al. assembled the MOIRE telescope [12], which consists of a 5 meter diameter membrane DOE primary lens that is combined with an optics assembly presenting an overall length of 27 m. By selecting National Imagery Interpretability Rating Scale (NIIRS) as criteria, the testing results demonstrated that a factor of 2.3 of NIIRS was achieved. Other types of diffractive telescopes, such as the photon sieve telescope [13] or the compound diffractive telescope [14], have been also proposed.

The above mentioned telescopes are very interesting proposals, but they also present certain disadvantages that cannot be neglected, which are mainly introduced by either small apertures or un-fully covered apertures. In order to eliminate such disadvantages, a larger aperture diffractive lens with a subaperture stitching structure is more desirable. However, the introduction of such lens in a diffractive telescope increases the space size of the telescope and leads to a large air turbulence effect [15]. Moreover, only the stitching errors of a single diffractive primary lens have been studied [16,17], but none stitching errors applied to diffractive telescopes has been studied so far.

In this framework, two different studies are provided in this paper with the idea to properly characterize and improve the performance of ground-based diffractive telescopes. First, the stitching error theory, recently used to characterize the performance of subaperture diffractive lenses [17], is applied to a subaperture diffractive lens contained telescope configuration. Note that we only focus on the analysis of the subaperture stitching error of the large diameter diffractive lens, other error sources such as the adjustment error among the optical elements in the telescope are not considered. Second, a ground-based Cassegrain diffractive telescope is proposed for the first time, which not only decreases the air turbulence problem significantly with higher resolution, but also leads to a more compact telescope scheme.

The outline of this paper is as follows. In Section 2, we study the performance of a subaperture stitching diffractive telescope in a transmissive configuration. In such configuration, the primary lens consists of a three-belt deployed subaperture stitching Fresnel lens [17], which can lead to stitching errors with five degrees of freedom (DOFs). An error analysis of the telescope shown in Section 2 is given in Section 3. Three types of stitching error are considered, being represented as a single stitching error and two comprehensive stitching errors within the transmissive subaperture telescope, respectively. In Section 4, a novel ground-based Cassegrain diffractive telescope design is proposed. This telescope has the same aperture as the transmissive configuration discussed in Section 2, but presents an enhanced performance and a more compact optical system. To highlight the improvement of the Cassegrain diffractive telescope, the characteristics of both the subaperture stitching telescope and the ground-based Cassegrain diffractive telescope are tested in Section 5 by analysing their particular star points and resolution board factors. Finally, the main conclusions of the work are given in Section 6.

## 2. Transmissive subaperture stitching diffractive telescope

The four main parts of a diffractive telescope system are the diffractive lens, the field lens, the Fresnel chromatic dispersion correction lens, and the converging lens. Figure 1 shows the layout of a diffractive telescope. When illuminating the optical scheme in Fig. 1 with a multi-wavelength collimated beam (represented in red), the light beams coming from the object disperses both spatially and spectrally (represented as the green, yellow and blue lines) after interacting with the diffractive lens. Afterwards, the field lens placed in a proper position focusses those dispersed beams to one point placed at the plane where the chromatic dispersion correction (CDC) lens is set (see Fig. 1). Note that the CDC lens introduced is a Fresnel lens correlated with the first diffractive lens. Next, the CDC lens, which is inversed to the first diffractive lens, eliminates the chromatic dispersion according to the Schupmann theory [6]. Finally, the last converging lens focusses the beam to the image plane.

Note that one of the most critical challenges in designing a diffractive telescope is to eliminate the chromatic aberration, so the Schupmann chromatic aberration correction theory [6] is considered first. The Schupmann theory demonstrates that any chromatic aberration originated by an optical element could be eliminated by introducing its inversing optical element at the image plane. By taking into account the Schupmann theory, the chromatic aberration of a diffractive telescope system, which arises from the two Fresnel lenses in its optical architecture (Diffractive lens and CDC lens in Fig. 1), can be eliminated if the two lenses satisfy the following condition [7],

where*f*is the focal length of the diffractive lens,

_{1}*f*is the focal length of the Fresnel CDC lens, and

_{3}*n*is a positive integer.

Note that the beam of the central wavelength should pass through the center of the field lens, so the distance between the diffractive lens and the field lens should be equal to the focal length of the diffractive lens working within the central wavelength. What is more, according to Schupmann theory, the diffractive lens and the Fresnel CDC lens are conjugated to the field lens.

By considering the above-discussed limitations when performing the optical design of the diffractive telescope, we implemented a 300 mm diameter stitching subaperture diffractive telescope of focal length 2000 mm, with a working wavelength range from 582.8 nm to 682.8 nm within a 0.15 degree field of view. The primary lens is a three-belt deployed Fresnel lens with a BK7 glass of 5 mm thickness, while the chromatic aberration correction Fresnel lens is a 50 mm diameter BK7 glass of 4 mm thickness. An aperture stop of 6.2 mm diameter is placed 34.330 m away from the diffractive lens to eliminate all the corresponding diffractive orders except for the + 1 order. The resulting total telescope track distance is 35.490 m. The angular resolution is of 0.00015°, which was obtained by considering the whole aperture with the central wavelength (632.8 nm).

The configuration of the stitching Fresnel lens was described previously [17] and is here reviewed. The centers for the three belts coincide with the center of the whole diffractive lens. The radius of the A-belt is 39.96 mm. The inner and outer radii for the B-belt are of 60.06 mm and 88.76 mm, respectively. Finally, the inner and outer radii for the C-belt are of 112.11mm and 149.99 mm.

After optimizing the telescope system, the Modulation Transfer Function (MTF) of the resulting telescope was simulated by using the Zemax software, and corresponding result is shown in Fig. 2. For simulations, three wavelengths are considered simultaneously: *λ*_{1} = 580 nm, *λ*_{2} = 633 nm, and *λ*_{3} = 680 nm.

In Fig. 2, we represent the MTF as a function of the spatial frequency with units of cycles per millimetre. The black curve in Fig. 2 represents the MTF distribution of the diffractive limit, and the remaining four curves represent the MTF of the diffractive limit, the 0° fields of view and the 0.075° fields of view in both sagittal (dashed lines) and tangential (continuous lines) directions. We see that by narrowing the field of view, better image quality is obtained, because higher spatial frequencies are not being filtered by the system.

Results in Fig. 2 provide that the subaperture stitching diffractive telescope could almost reach the diffraction limit (black line) in 0° field of view, and the spatial frequency within this field of view could reach 80 cy/mm with a modulation factor of 0.27 in tangential direction. What is more, the modulation factor reaches 0.1 with a 78 cy/mm spatial frequency in the edge field of view. The maximum telescope chromatic aberration reaches 0.2λ between the central wavelength (633nm) and the edge wavelength (580nm) in 0° field of view and 3λ in the sagittal direction in 0.075° field of view by taking consideration of the OPDs. The simulation result shows a weak chromatic aberration, which can demonstrate the correctness of Schupmann theory introduced in the design of the diffractive telescope. Note that weak chromatic aberrations hardly influence the image quality, and thus, the chromatic aberrations are not discussed in the following simulations.

## 3. Error analysis of subaperture stitching diffractive telescopes

Most super-large diameter telescopes nowadays implemented are based on subaperture stitching primary mirrors [15, 18–22]. Therefore, the study of stitching errors within subaperture diffractive telescopes becomes mandatory. However, there is a lack in literature of methods for the estimation of stitching errors in diffractive telescopes. Hence, we use the stitching error tolerance theory that we recently provided [17] to fulfil the analysis. This method, which is based on the Rayleigh Criterion, proved to be suitable to estimate the stitching tolerances of diffractive lenses. In this section, we apply it to investigate the stitching error tolerances in diffractive telescopes. As it is discussed, the subaperture diffractive lens can be modelled as a lens with five degrees of freedom (DOFs) stitching errors, and these errors can be compacted in three different types. These three different types of error tolerance can be calculated, by using the following three relations, which are deduced by selecting a PV of λ/10 according to the Rayleigh Criterion [17],

The parameter Δ*r* represents the radial offset errors tolerance (Eq. (2)), with λ being the wavelength of the illumination, and *f ^{#}* corresponds to the f-number of the stitching diffractive lens. The parameters Δ

*z*(Eq. (3)) and Δ

*θ*(Eq. (4)) describe the axial offset error tolerance and the rotating error tolerance, respectively. More geometrical details of remaining offset errors can be found in Figs. 1-3 of [17]. In Eq. (4),

*r*is the radius of a single subaperture, and

*d*is the distance from the geometric center of the subaperture to the whole diffractive lens center. Finally, the parameter Δ in Eq. (4) is calculated as

By using the collection of Eqs. (2)-(4), the Δ*r,* Δ*z* and Δ*θ* errors tolerances were calculated for each one of the three belts. In all the cases, the central wavelength *λ* = 633 nm was selected and the particular values for the parameters *r* and *f*^{#} were considered for each particular belt. The corresponding simulated results are given in Table 1.

Note that in Fig. 2 we simulated the performance of the transmissive telescope in terms of MTF without considering the effect of stitching errors. By taking into account the above-stated discussion, the same simulation was repeated, but now, by considering the errors tolerances given in Table 1. In the following, we perform three different simulations to evaluate the stitching error in the performance of the transmissive telescope, the single stitching error (Sec. 3.1), and the overall stitching errors (Sec. 3.2 and Sec. 3.3) cases.

#### 3.1 Subaperture stitching telescope with a single stitching error

The influence of a single stitching error on the performance of the transmissive telescope sketched in Fig. 1 is here studied. For simulations, an axial offset error at the C-belt is considered in detail first, the rest offset errors in each belt are studied with the same method afterwards. According to Eq. (3), and the particular parameters for the C-belt, the axial offset error tolerance Δ*z* was found to be of 5.06 mm (see Table 1). To test the validity of the tolerance calculation above-proposed, we analyse the bounds of the telescope performance, by studying the effect of an axial error value slightly below or above this estimated axial offset error tolerance of 5.06 mm. We begin by selecting for simulations an axial offset error slightly below the error tolerance, in particular, with a value of 5 mm. By using this value, the MTF of the corresponding telescope was simulated by using Zemax software. Obtained results are shown in Fig. 3, where different colours curves represent different fields of view. Note that continuous and dashed lines represent tangential and sagittal directions, respectively.

By comparing the MTF curves for different fields of view without considering stitching errors (Fig. 2) with those calculated by considering an axial offset error Δ*z* close to the estimated axial tolerance value (5 mm, see Fig. 3), we see that they lead to practically equivalent results. For instance, we calculated the absolute MTF differences (MTF_{without stitching error} – MTF_{with stitching error}) between the edge field of view within tangential direction (square curves) in Figs. 2 and 3. The corresponding MTF differences are plotted as squares (Dif. 1) in Fig. 4. To the sake of visualization, only high spatial frequencies are shown in Fig. 4 (from 75 to 80 cy/mm), as they are more significant in terms of image quality. However, the same behaviour is observed at the full spatial frequency space. Note that Dif. 1 curve almost reaches zero in each spatial frequency, with the values being permanently lower than 0.0075. Therefore, data in Fig. 4 demonstrates that the image quality is not significantly reduced if a single stitching error is considered, as long as the stitching error values range within the telescope error tolerance (Table 1).

Secondly, we analyse the upper bound, i.e., axial offset errors larger than those of the telescope tolerance error. In this case, a simulation of the telescope performance with a 7 mm axial offset error in the C-belt was also performed and the corresponding results are given in Fig. 5. In such a case, the MTF curves show a modified behaviour as a function of the spatial frequency. For instance, a modulation factor of 0.1 in the edge field of view is obtained for 78 cy/mm without stitching errors (0.075° tangential curve in Fig. 2), but it drops to 74 cy/mm by including the 7 mm axial error (Fig. 5), this being a behaviour more abruptly within the same field of view.

The absolute MTF difference values between the 7 mm radial-offset-error telescope and the non-error-contained telescope were calculated as well. Results are provided as circles (Dif. 2) in Fig. 4. Note that Dif. 2 curve provides MTF differences values nearly two times (0.014) those of Dif. 1 in each spatial frequency selected, providing a significantly different performance between telescopes. Thus, the optical image quality is reduced by including a stitching error bigger than its error tolerance.

The above-stated simulation of the C-belt axial offset error provides a method to evaluate single stitching error contained telescopes completely and thoroughly. Here, we generalized the study by applying this method to the rest of offset errors in the whole system of belts. The corresponding results are simulated and presented in Tab. 2. For the new simulations, we used a criterion equivalent to that used for the above-discussed C-belt analysis, i.e., for each new error tolerance evaluated, we selected error values slightly above the calculated error tolerances. The same factor of 1.4 (above the error tolerance) selected in the previous analysis, is also used here.

Table 2 provides the spatial frequency values of the single stitching error within all belts for the Δ*z*, Δ*θ* and Δ*r* offset cases. For instance, the second and the third columns in Tab. 2 represent the single axial stitching offset provided in the telescope and the corresponding spatial frequency achieved by selecting a modulus factor of 0.1 as criterion. Moreover, each row represents the corresponding belt in the subaperture diffractive lens. We want to emphasize that each grouped error offset and spatial frequency is obtained when only one single error is generated in the telescope. We see that by selecting the same proportioned offset error (the 1.4 factor) used in the previous single C-belt study, we obtain almost the same behaviour of the system MTF as a function of the spatial frequency. In particular, for all the offset errors evaluated within all the belts, we observe a decrease in the spatial frequencies from 80 cy/mm to a range within 74-77 cy/mm. This result demonstrates that different single stitching offsets provide a nearly equal sensitivity at different regions of the diffractive lens.

Note that the above-stated discussion validates the stitching error tolerance theory, and thus, this approach is suitable for the evaluation of subaperture stitching diffractive telescopes. In fact, tolerances calculated with Eqs. (2)-(4) properly set the thresholds estimating the error ranges where the quality of images is not significantly decreased by stitching errors.

#### 3.2 Subaperture stitching telescope with a comprehensive stitching error in the C-belt

In real implementations, instead of only one type of stitching error, a diffractive lens contains an exhaustive stitching error that includes both the rotating error and the offset (axial and radial) errors. What is more, according to the overall error theory [17], the rotating error and the axial offset error are more sensitive, in terms of image quality, comparing to the radial offset error. In this situation, if a particular telescope tolerance is fixed, the values for each particular error constituting the comprehensive error must be smaller than the value of a single error leading to the same tolerance. Hence, the overall error can be calculated as Eq. (6) [17],

*L*represents the criterion of the overall errors (the Rayleigh Criterion, by selecting PV of λ/10),

*Δl*are the effects of individual errors on the final image quality, and

_{i}*W*are the weights of each error (

_{i}*W*and

_{1}*W*for rotating errors in X and Y direction,

_{2}*W*for the axial offset error, and

_{3}*W*and

_{4}*W*for radial offset errors in X and Y direction). We want to note that if different weight combinations were considered, different results would be obtained in terms of the image quality. However, as long as the exhaustive error is ranging within the Rayleigh Criterion, the variations in the image quality are negligible.

_{5}In order to mimic a real implementation, we set the rotating error weights of W* _{1}* = W

*= 0.67, the axial offset error weight of W*

_{2}*= 0.67, the radial offset error weights of W*

_{3}*= W*

_{4}*= 0.60, and the Rayleigh Criterion 0.1λ. Note that the proportion (the weight) of each error we used to construct the overall error is approximately equivalent, this being a common situation in real implementations. The corresponding results are provided in Table 3.*

_{5}Again, the telescope performance was studied in terms of MTF. In this case, only the comprehensive error was considered for C-belt (we assume that the other two belts are ideally stitched). However, the same method described in the following could be also applied for the A and B belts, or in a more general way, to other subaperture diffractive lens shapes.

Analogous to the single stitching error study in Sec. 3.1, in this case, we simulate overall errors slightly below and above the calculated tolerance in Table 3. In the case of error values slightly below the tolerances, we set: 7.1 μm for the radial offset (both tangential and sagittal directions), 3.3 mm for the axial offset, and 0.045° for the rotation offset (both tangential and sagittal directions). The simulation results show practically equivalent MTF curves than those in Fig. 2 (without considering stitching errors). To reinforce this statement, the absolute MTF differences between an exhaustive errors contained telescope (within the error tolerance) and a none-error-contained telescope were calculated for the 0.075° field of view case. Corresponding data are provided as triangles (Dif. 3) in Fig. 6. For all the spatial frequencies, the difference values are all smaller than 0.01. Note that this result is in the same order as that obtained for the single stitching error study of Sec. 3.1 (see Fig. 4). Therefore, a stitching error within its comprehensive error tolerance does not considerably degrade the image quality of a diffractive telescope.

We also analysed overall errors beyond the tolerances in Table 2. In this case, we set the following overall errors: 8 μm for the radial offset (both tangential and sagittal directions), 3.5 mm for the axial offset, and 0.06° for the rotation offset (tangential and sagittal directions). The corresponding MTF difference curve is given as diamonds (Dif. 4) in Fig. 6, and they are essentially above 0.012, which represents a significant increase when compared with Dif. 3. Note that the data in Fig. 6 behave similarly to that in Fig. 4, and thus, Eq. (6) properly allows setting a threshold for exhaustive error tolerances. Under this scenario, the optical image quality is reduced by including a stitching comprehensive error bigger than its error tolerance.

#### 3.3 Subaperture stitching telescope with a comprehensive error in all the belts

Typically, to achieve a large diameter for the primary diffractive lens of a telescope, it is compulsory to manufacture numbers of subapertures and stitch them together, where each one presents their own comprehensive errors. When compared with other methods, the stitching approximation allows that the whole aperture can be fabricated avoiding important technical difficulties and reducing industrial costs. Under this scenario, it is mandatory to study whether each subaperture is influencing the image quality separately or synthetically.

To this aim, we simulated the telescope performance in terms of MTF when considering comprehensive errors in each belt (A, B and C belts). In this case, we used overall errors slightly below the tolerance values previously provided in Table 3. The particular values used for each belt are given in Table 4.

The corresponding MTF of the telescope containing the above-stated comprehensive errors is given in Fig. 7.

For the sake of clarity, we focus on the discussion in the same field of view than in previous analysis (0.075° in tangential direction). The corresponding MTF is barely changed compared to the curves in Fig. 5 (overall errors only in C-belt). This can be seen by calculating the absolute differences between the MTF for the single overall error curve (Fig. 5) and the corresponding one for the three-belts overall error curve in Fig. 7. Corresponding results are given in Fig. 8 as stars (Dif. 5). For visual comparison, the MTF difference between the ideal stitched telescope (Fig. 2) and the C-belt overall error telescope (Fig. 5), which was already represented in Fig. 6 (Dif. 3), is also included in Fig. 8. By comparing Dif. 3 and Dif. 5 curves in Fig. 8, we see they have a very similar behaviour for all the spatial frequencies, with very small differences (∼0.00075). This result ensures that the performance of the telescope with exhaustive errors only at the C-belt, and the telescope with errors at the three belts, is almost equivalent in terms of MTF. As a consequence, we demonstrated that, not synthetically, but each subaperture introduced in a diffractive telescope influences the image quality separately. As shown in the following, this statement is reinforced by also considering the A-belt and the B-belt cases. In particular, we also studied the MTF differences of the A-belt and the B-belt exhaustive error contained telescope with the ideally stitched telescope, respectively. The corresponding results are shown as Dif. 6 (A-belt error case) and Dif. 7 (B-belt error case) in Fig. 8. Note that for all the cases, MTF differences are nearly equivalent to the maximum MTF difference obtained for the exhaustive error case (Dif. 5). This result supports the conclusion that the subapertures are influencing the image quality separately and the combined effect is negligible within the telescope error tolerances.

## 4. Ground-based Cassegrain diffractive telescope

So far, a method to evaluate error tolerances in diffractive telescopes was studied and tested. In this section, we propose to implement a ground-based diffractive telescope based on a Cassegrain optical scheme, because this leads to a better telescope performance, as provided in the following discussion. In particular, the proposed Cassegrain diffractive telescope led to a compact optical system that reduces the optical track. This scheme is more favourable in the ground-based test since it reduces the turbulence impact, leading to an improvement of the image contrast when compared with tranmissive configurations.

The basic implementation of a ground-based Cassegrian diffractive telescope mainly shares the architecture of the transmissive telescope shown in Fig. 1. A sketch of the Cassegrian diffractive telescope is pictured in Fig. 9. The primary diffractive lens (see green arrow in Fig. 9) is again a diffractive lens except for the fact that an aspherical mirror is placed in the center of the lens (see orange arrow in Fig. 9). Note that this diffractive lens consists of a single glass piece with a hole at the center, avoiding the stitching errors present at the diffractive transmissive configuration. An aspherical or spherical mirror with a transmission aperture in the center (see dashed blue arrow in Fig. 9), labelled as primary mirror, is placed to a certain distance from the primary lens to construct the Cassegrian conformation with the reflective mirror on the diffractive lens. Note that even though the Cassegrian implementation introduces no chromatic aberration, the chromatic dispersion correction lens and the primary diffractive lens should still obey the Schupmann principle, because the primary diffractive lens introduces the chromatic aberration. Finally, a converging lens is used to collect the image in the camera.

Even though the collimated light disperses after the primary diffractive lens, the combination of the primary mirror and the secondary mirror reduces the dispersion significantly because they provide a more compactable optical path compared to the transmissive telescope. The converging light reflected from the primary mirror is reflected for a second time at the secondary mirror to meet the chromatic dispersion correction lens. Note that the two-mirror-based Cassegrian structure introduces no chromatic aberration. Moreover, because of the chromatic correction lens and the primary diffractive lens fit the Schupmann principle, the light shows no chromatic aberration when reaching the converging lens.

In Sections 2 and 3 we studied a 300 mm diameter transmissive telescope. To perform a proper comparison with the transmissive telecsope, we implemented a Cassegrain diffractive telescope with as well 300 mm diameter and a total track distance of 1.267 m, working in the wavelength range from 582.8 nm to 682.8 nm. The focal length of the telescope was of 1502 mm, with the primary lens being an integral diffractive lens of 300 mm diameter. This diffractive lens contains a 64 mm diameter circular spherical mirror in the center. The primary reflective mirror was located 0.708 m from the diffractive lens. By selecting 80 cy/mm spatial frequency at the edge field of view as the criterion, the MTF simulations for the one-piece and for the 3-belt subaperture diffractive lens contained transmissive telescopes shown a likely equivalent image quality. By taking into account that similarity observed at the simulated results, we can use the 3-belt subaperture contained telescope to fulfil the comparison with the one-piece lens based Cassegrain configuration.

For comparison, we simulated the MTF of the above-proposed Cassegrain telescope, and the obtained results are given in Fig. 10. By comparing results in Fig. 10 with those of the transmissive configuration (Fig. 2), we see that the Cassegrain based configuration leads to an improvement of the image quality. In particular, the diffractive limit could reach 80 cy/mm with a modulation factor 0.41 (black curve), while in the edge field of view, the spatial frequency reaches 80 cy/mm with a modulation factor 0.36 (0.075° curve). These results significantly improve the MTF compared to Fig. 2, proving that the image quality is refined. We want to note that it is the large effective aperture of the Cassegrain telescope provides the improvement of the image quality. More importantly, according to the experimental test that is provided in the following section (Sec. 5), the ground-based Cassegrain diffractive telescope shows a higher contrast resolution board and less sensibility to air turbulence. In particular, the compact Cassegrain diffractive telescope designed reduces the optical track from 35.490 m to 1.267 m, when compared with the transmissive configuration in Section 2.

## 5. Evaluation of the proposed telescopes: experimental results

Further comparisons between the transmissive subaperture diffractive telescope and the Cassegrian diffractive telescope are provided in this section to demonstrate the correctness of the stitching error tolerance theory as well as the effectiveness of the proposed Cassegrain scheme. To this aim, the well-known star point and the resolution board tests were performed on the experimental implementations of both the transmissive telescope and the Cassegrian telescope under the same experimental conditions (i.e., illumination, temperature, air turbulence, etc.). Note that except for the tests in the performance of the Cassegrain telescope, the transmissive telescope was also tested by both introducing different stitching errors and not introducing any stitching error, respectively. In particular, the tests for the ideally stitched transmissive telescope were conducted without artificially introducing any stitching error in the three subapertures in the diffractive lens. By contrast, the effect of stitching errors was included by artificially implementing different stitching errors in the transmissive diffractive telescope. Note that in the case of the Cassegrian configuration, due to its optical scheme, no stitching errors are presented.

The implemented experimental transmissive and Cassegrain telescopes are shown in Fig. 11. In the case of the transmissive telescope (Fig. 11(a)), the telescope is based on two main sections (labelled as Sections 1 and 2 in Fig. 11(a)). In Sect. 1, we can see a detail of the illumination system and the first diffractive lens, while in Section 2, we show the rest of the elements in the telescope system. The illumination (632.8nm) from the He-Ne laser is firstly expanded and filtered by the pinhole (5µm diameter, situated at the focal plane of the microscope) and the microscope. Afterwards, the collimated beam obtained from the collimator is propagated to meet the first diffractive lens. The final images are received by introducing a CCD to the focal plane of the telescope system. Note that the optical sketch of the subaperture telescope is the same to the one-piece diffractive lens transmissive telescope. Then, just by replacing the one-piece lens (indicated by a blue line bar) by the subaperture stitching lens, we can obtain the other configuration. On the other hand, a detail of the Cassagrian telescope is shown in Fig. 11(b). The primary mirror fixed to the mechanical support structure demonstrates a diameter of 298.6 mm. What is more, the CDC lens system, as well as the converging lenses (the focusing system) is calibrated together with the CCD detector. Note that the secondary mirror is missing in Fig. 11(b) for the sake of visibility.

#### 5.1 Star point test

Note that the point spread function (PSF) of the system is tested, describing the response of an imaging system to a plane wavefront. The 300 mm diameter subaperture transmissive telescope and the ground-based Cassegrain diffractive telescope were tested under the same experimental condition.

In this first experiment, no stitching error is considered (ideal transmissive telescope). The testing results are shown in Fig. 12. In Fig. 12, we achieve the well-known Airy pattern related to the PSF of circular apertures. Whereas Fig. 12(a) shows the image obtained at the CCD camera when using the transmissive sub-aperture telescope, Fig. 12(b) gives the corresponding image for the Cassegrian telescope case. The imaged star points in Figs. 12(a) and 12(b) demonstrate that both telescopes can provide a good image quality. What is more, the ground-based Cassegrain diffractive telescope (Fig. 12(b)) leads to a higher PSF quality, with a better clearness and uniformity than that of the transmissive telescope. This improvement is related to the character of the Cassegrain scheme as well as the optical system track.

The same star-based experiment was repeated for the transmissive diffractive telescope, but now by considering stitching errors. In this way, the simulated error tolerances analysis provided in Section 3 are here experimentally verified. The corresponding PSFs for those cases are provided in Fig. 13. In particular, four different situations are considered: a single axial offset error of 5 mm and 7 mm in the C-belt (Figs. 13(a) and 13(b), respectively); an exhaustive stitching error tolerance (data in Table 3) within the C-belt (Fig. 13(c)); and the overall error tolerances (data in Table 4) within the A, B and C belts (Fig. 13(d)).

Image given in Fig. 13(a) represents a uniform star point quite similar to that in Fig. 12(a), which demonstrates that the image quality is not significantly reduced if the stitching error ranges within the error tolerance. However, image in Fig. 13(b) is more distorted and presents an obvious reduction of uniformity. This implies that when the stitching error exceeds the error tolerance of the telescope, a reduction of the image quality is obtained. Note that experimental results in Figs. 13(a) and 13(b) fully agree with the error tolerance discussion provided in Section 3. Moreover, the similarity of the two imaged star points in Figs. 13(c) and 13(d), obtained by using two different overall errors (single exhaustive error in C-belt and exhaustive errors in A, B and C belts), is in agreement with the discussion provided in Section 3.3. In fact, we experimentally verify that exhaustive errors influence the telescope image quality separately, but not synthetically.

#### 5.2 Resolution board test

The resolution board test was also performed on the two telescopes. In this case, we used a bright LED with its wavelength ranging from 620 nm to 640 nm as the illumination. The resolution board test results of the ground-based Cassegrian telescope and the subaperture transmissive telescope are provided in Figs. 14 and 15.

Figure 14 reveals the resolution board tests of the ideally stitched subaperture transmissive telescope (without considering any stitching error) (Fig. 14(a)) and the ground-based Cassegrain telescope (Fig. 14(b)). The ideally stitched transmissive telescope reached its image limitation to a resolution of 75 cy/mm. Note that in the experimental implementation, not only the air turbulence blurs the image contrast (see Fig. 14(a)), but also provides a smaller resolution than the simulation results in Sec. 2 (i.e., simulations reached 78 cy/mm). On the contrary, an 81 cy/mm resolution provided by the Cassegrain telescope referred in Fig. 14(b) offers a higher resolution compared to the transmissive one. What is more, its high contrast and clearness proved that the compacted Cassegrain diffractive telescope can decrease the influence of the air turbulence to a significant extent in the ground-based test.

Afterward, the resolution boards of the subaperture transmissive telescope containing four kinds of stitching error were tested, respectively. In order to obtain a quantitative analysis of the stitching error in the transmissive telescope, the same four stitching errors set for the star test analysis in Sec. 5.1, were also used for the board test. The resolution boards of the transmissive telescope carrying the above-stated four errors are shown in Fig. 15.

Figure 15(a) shows as 75 cy/mm are resolved, showing the maximum resolution of the telescope when considering a 5 mm axial offset stitching error in C-belt. In the case an offset error increased to 7mm, the resolution is decreased to 71 cy/mm (Fig. 15(b)). Therefore, the exceed of the error tolerance in a 300 mm aperture transmissive stitching telescope introduces a reduction of the resolution from 75 cy/mm to 71 cy/mm.

In the case of comprehensive errors, we artificially set the errors in Table 3, for the case of a transmissive telescope containing an overall error tolerance in C-belt, and errors in Table 4, for the telescope accommodating three exhaustive error tolerances in all belts. Obtained results are given in Figs. 15(c) and 15(d). We see that the 75 cy/mm resolution achieved by the transmissive telescope containing an overall error tolerance in C-belt (Fig. 15(c)), coincides with the resolution, 75 cy/mm, of the transmissive telescope accommodating three exhaustive error tolerances in all belts (Fig. 15(d)). The resolution board tests verified the stitching error theory applied to the subaperture stitching telescope in Sec. 3, quantitatively.

## 6. Conclusion

In this paper, two main contributions are proposed and discussed. On the one hand, a ground-based Cassegrain scheme is proposed for the implementation of diffractive telescopes. The 300 mm diameter prototype not only provides a more compact structure, but also presents a better image quality compared to the transmissive diffractive telescope working under the same conditions. An 81 cy/mm resolution is easily achieved in the 1.502 m track Cassegrain diffractive telescope compared to the 75 cy/mm resolution with low contrast within the 35.490 m transmissive diffractive telescope. On the other hand, we also verified the effectiveness of applying the stitching error theory to a subaperture diffractive telescope. Simulations within the three-belt distributed subaperture stitching diffractive telescope by considering both the single stitching error and the comprehensive stitching error are provided. In addition, we experimentally implemented both the Cassegrain and the transmissive diffractive telescopes, and the performance of the telescopes was experimentally evaluated.

We want to note that the ground-based Cassegrain diffractive telescope not only provides a better solution to realize the tests on the ground, but also affords the possibility to fulfil a more simple space alignment in the future astronomical applications due to it offers a more compacted configuration. Apart from the Cassegrain scheme, the stitching error tolerances investigated, as well as the conclusion that independent subapertures in a diffractive telescope influence the image quality separately, provide the instruction of the subaperture design and deployment for super-large diameter primary lenses used in the space diffractive telescopes. Further investigation is required on large diameter subaperture stitching diffractive lens testing method and the alignment technique for both the subaperture diffractive lens and the complete telescope system of a meter-size space diffractive telescope.

## Funding

Jilin Province Youth Leading Talent and Team Project (20160519021JH).

## Acknowledgments

This work is supported by the Innovation of Science and Technology in Jilin Province Youth Leading Talent and Team Project and State Key Laboratory of Luminescence and Applications.

## References and links

**1. **R. Hyde, S. Dixit, A. Weisberg, and M. Rushford, “Eyeglass, A very large aperture diffractive space telescope,” Proc. SPIE **4849**, 28–39 (2002).

**2. **E. Hinglais, “A space Fresnel imager concept assessment study led by CNES for astrophysics applications,” Exp. Astron. **30**(2–3), 85–110 (2011).

**3. **P. Deba, P. Etcheto, and P. Duchon, “Preparing the way to space borne Fresnel,” Exp. Astron. **30**(2–3), 123–136 (2011).

**4. **T. Hawarden, C. Johnstone, and G. Johnstone, “GISMO, an ELT in space: a giant (30-m) far-infrared and submillimeter space observatory,” Proc. SPIE **5382**, 95–104 (2003).

**5. **D. A. Buralli and G. M. Morris, “Design of two- and three-element diffractive Keplerian telescopes,” Appl. Opt. **31**(1), 38–43 (1992). [PubMed]

**6. **L. Schupmann, *Die Medial-Fernrohre - Eine neue Konstruktion für große astronomische Instrumente*, (Tuebner-Verlag, 1899).

**7. **R. A. Hyde, “Eyeglass. 1. Very large aperture diffractive telescopes,” Appl. Opt. **38**(19), 4198–4212 (1999). [PubMed]

**8. **G. K. Skinner, “Design and imaging performance of achromatic diffractive-refractive X-ray and Gamma-ray Fresnel lenses,” Appl. Opt. **43**(25), 4845–4853 (2004). [PubMed]

**9. **G. Anderson, O. Asmolova, and T. Dickinson, “FalconSAT-7: a membrane space telescope,” Proc. SPIE **9085**, 908504 (2014).

**10. **L. Koechlin, J. Rivet, P. Deba, D. Serre, T. Raksasataya, R. Gili, and J. David, “First high dynamic range and high resolution images of the sky obtained with a diffractive Fresnel array telescope,” Exp. Astron. **33**(1), 129–140 (2012).

**11. **L. Koechlin, M. Yadallee, T. Raksasataya, and A. Berdeu, “New progress on the Fresnel imager for UV space astronomy,” Exp. Astron. **354**(1), 147–153 (2014).

**12. **P. Atcheson, C. Stewart, J. Domber, K. Whiteaker, J. Cole, P. Spuhler, and A. Seltzer, “MOIRE- Initial demonstration of a transmissive diffractive membrane optic for large lightweight optical telescope,” Proc. SPIE **8442**, 844221 (2014).

**13. **G. Andersen and D. Tullson, “Broadband antihole photon sieve telescope,” Appl. Opt. **46**(18), 3706–3708 (2007). [PubMed]

**14. **H. Liu, Z. Lu, J. Yue, and H. Zhang, “The characteristics of compound diffractive telescope,” Opt. Express **16**(20), 16195–16201 (2008). [PubMed]

**15. **J. Domber, P. Atcheson, and J. Kommers, “MOIRE: ground test bed results for a large membrane telescope,” in *AIAA SciTech Space Structures Conference* (2014), pp. 1510.

**16. **G. Jin, J. Yan, H. Liu, X. Zhong, and Y. Yan, “Flat-Stitching error analysis of large-aperture photon sieves,” Appl. Opt. **53**(1), 90–95 (2014). [PubMed]

**17. **H. Zhang, H. Liu, W. Xu, A. Lizana, X. Wang, and Z. Lu, “Error analysis of large diameter subaperture stitching Fresnel diffractive lens,” Appl. Opt. **56**(27), 7672–7678 (2017).

**18. **J. Gardner, J. Mather, M. Clampin, R. Doyon, M. Greenhouse, H. Hammel, J. Hutchings, P. Jakobsen, S. Lilly, K. Long, J. Lunine, M. McCaughrean, M. Mountain, J. Nella, G. Rieke, M. Rieke, H. Rix, E. Smith, G. Sonneborn, M. Stiavelli, H. Stockman, R. Windhorst, and G. Wright, “The James Webb Space Telescope,” Space Sci. Rev. **123**(4), 485–606 (2006).

**19. **J. Gardner, J. Mather, M. Clampin, R. Doyon, K. Flanagan, M. Franx, M. Greenhouse, H. Hammel, J. Hutchings, P. Jakobsen, S. Lilly, J. Luinine, M. McCaughrean, M. Mountian, G. Rieke, M. Rieke, G. Sonneborn, M. Stiavelli, R. Windhorst, and G. Wright, “The James Webb Space Telescope,” in *Astrophysics in the Next Decade*, H. Thronson, M. Stiavelli, and A. Tielens (Springer, 2009).

**20. **G. Olczak, D. Fischer, M. Connelly, and C. Wells, “James Webb Space Telescope primary mirror integration: testing the multiwavelength interferometer on the test bed telescope,” Proc. SPIE **8146**, 814608 (2011).

**21. **B. Saif, D. Chaney, W. Smith, P. Greenfield, W. Hack, J. Bluth, A. Van Otten, M. Bluth, J. Sanders, R. Keski-Kuba, L. Feinburg, M. North-Morris, and J. Millerd, “Nanometer level characterization of the James Webb Space Telescope optomechanical systems using high-speed interferometry,” Appl. Opt. **54**(13), 4285–4298 (2015).

**22. **G. Matthews, T. Whitman, L. Feinberg, M. Voyton, J. Lander, and R. Keski-Kuba, “JWST telescope integration and test progress,” Proc. SPIE **9904**, 990404 (2016).