Experimental investigation of binodal astigmatism in nodal aberration theory (NAT) with a Cassegrain telescope system

Özgür Karcı; Eray Arpa; Mustafa Ekinci; Jannick P. Rolland

doi:10.1364/OE.427164

1. Introduction

Nodal aberration theory (NAT) describes the aberration properties of optical systems without symmetry, initially driven by the need to understand and quantify the aberrations of misalignments. The roots of NAT can be traced to a “through-focus star plate” image that was taken by the 60’’ telescope located on Kitt Peak in the mid-70s showing binodal astigmatism and brought to the attention of Roland V. Shack [1]. The star plate image was taken prior to theoretical developments that led to NAT's discovery by Shack [2]. The theory is based on the wave aberration theory of H. H. Hopkins and the concept of shifted aberration field centers first developed by R. Buchroeder [3,4]. NAT was fully developed up to fifth-order by Kevin P. Thompson [5–8]. Recently, NAT was expanded through the aberrations induced by freeform surfaces [9–11], which shows the significance of the theory.

Despite the success and well-established theoretical developments of NAT, an experimental investigation is still in progress. Recently, Zhao et al. experimentally validated third-order coma in NAT for a customized Ritchey-Chrétien (RC) telescope system without symmetry (i.e., intentionally misaligned). Results demonstrate field-constant induced coma from misalignments [12]. Experimental validation of binodal astigmatism, which is the root of NAT, has been challenging.

This study presents the simulation and experimental validation of binodal astigmatism in NAT with a customized, high-precision Cassegrain telescope system designed and assembled by Karci and Ekinci [13]. The telescope uses high-precision flexure stages to intentionally introduce the secondary mirror's misalignments to generate aberrations of misalignment. For experimental validation of binodal astigmatism, the coma-free pivot point was simulated, and the binodal astigmatic field response was isolated. As background, this research was planned and started as a consecutive study of Zhao et al. on the coma-corrected RC telescope, which is an ideal system to show binodal astigmatism in NAT. The experiments were interrupted because of historical reasons (i.e., the COVID-19 pandemic). Furthermore, the RC telescope displays a mount-induced figure error in the primary mirror, which is creating complexity in decoupling alignment from mount-induced astigmatism. Consequently, the simulations performed for the RC telescope were adapted to an existing Cassegrain telescope located in Turkey, and this study was completed successfully. The RC telescope project remains in progress.

The study is organized as follows. Section 2 provides the theoretical background of third-order astigmatism and the coma-free pivot point. Section 3 demonstrates a Cassegrain telescope system, and Section 4 presents the simulation results and experimental setup. Section 5 provides experimental results and discussions.

2. Nodal property of third-order astigmatism in a misaligned optical system

The vector form of the wave aberration expansions can be written for third-order aberrations in an optical system without symmetry (i.e., misaligned or intentionally decentered/tilted) as

(1)$$\begin{aligned} \textrm{W} &= \mathop \sum \limits_j {W_{040j}}{({{\boldsymbol \rho }.{\boldsymbol \rho }} )^2} + \mathop \sum \limits_j {W_{131j}}[{({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} ).{\boldsymbol \rho }} ]+ \mathop \sum \limits_j {W_{222j}}{[{({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} ).{\boldsymbol \rho }} ]^2}\\ &\quad + \mathop \sum \limits_j {W_{220Sj}}[{({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} ).({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} )} ]({{\boldsymbol \rho }.{\boldsymbol \rho }} )\\ &\quad + \mathop \sum \limits_j {W_{311j}}[{({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} ).({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} )} ][{({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} ).{\boldsymbol \rho }} ]\; \; \; , \end{aligned}$$

where H denotes the normalized vector of the field coordinate on the image plane, ρ denotes the normalized vector of the pupil coordinate on the exit pupil plane, and σ_j denotes the displacement in the center of the aberration field associated with the surface j concerning the unperturbed field center [14,15]. The first, second, third, fourth, and fifth summations designate third-order spherical aberration, coma, astigmatism, field curvature, and distortion, respectively. W_222j represents the wave aberration term contribution for third-order astigmatism of surface j. In rotationally symmetric optical systems, astigmatism is defined with quadratic field and pupil dependence. The field behavior of astigmatism with respect to the medial surface is expressed in the presence of misalignments as

(2)$$\begin{aligned} W &= \frac{1}{2}\mathop \sum \limits_j {W_{222j}}{[{({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} ).{\boldsymbol \rho }} ]^2}\\ &= \frac{1}{2}\left[ {\mathop \sum \limits_j {W_{222j}}{{\boldsymbol H}^2} - 2{\boldsymbol H}\left( {\mathop \sum \limits_j {W_{222j}}{{\boldsymbol \sigma }_j}} \right) + \mathop \sum \limits_j {W_{222j}}{{\boldsymbol \sigma }_j}^2} \right].{{\boldsymbol \rho }^2}{\; }. \end{aligned}$$

This expression may be simplified by defining two unnormalized displacement vectors that define the sum of the surface contribution displacement vectors in the image plane, each weighted by the corresponding surface contribution to third-order astigmatism [5] as

(3)$${{\boldsymbol A}_{222}} \equiv \mathop \sum \limits_j {W_{222j}}\; {{\boldsymbol \sigma }_j}\; $$

(4)$${\boldsymbol B}_{222}^2 \equiv \mathop \sum \limits_j {W_{222j}}\; {{\boldsymbol \sigma }_j}^2\; \; \; \; ,$$

and the normalized vectors, a₂₂₂ and b₂₂₂, provided W₂₂₂ ≠0, can be expressed as:

(5)$${{\boldsymbol a}_{222}} \equiv \frac{{{{\boldsymbol A}_{222}}}}{{{W_{222}}}}\; \; $$

(6)$${\boldsymbol b}_{222}^2 \equiv \frac{{{\boldsymbol B}_{222}^2}}{{{W_{222}}}}\; - {\boldsymbol a}_{222}^2,\; $$

where W₂₂₂, a₂₂₂, and b₂₂₂ are the total wave aberration for astigmatism, a vector from the center of the field to the midpoint between two astigmatic nodes, and two vectors pointing from the endpoint of the a₂₂₂ vector to the two astigmatic nodes, respectively. This new characteristic field dependence for astigmatism in the optical system without symmetry can now be expressed as

(7)$$W = \frac{1}{2}{W_{222}}[{{{({{\boldsymbol H} - {{\boldsymbol a}_{222}}} )}^2} + {\boldsymbol b}_{222}^2} ].{\; }{{\boldsymbol \rho }^2} = {\; }\frac{1}{2}{W_{222}}[{{\boldsymbol H}_{222}^2 + {\boldsymbol b}_{222}^2} ].{\; }{{\boldsymbol \rho }^2}{\; \; }.$$

Equation (7) includes terms that are squared vectors. They represent vector multiplication, which is not a commonly used operation, and discussed extensively by Thompson [5], Appendix A. Equation (7) can be solved for locations where the astigmatic terms go to zero for H as [5]

(8)$${\boldsymbol H} = {{\boldsymbol a}_{222}} \pm i{{\boldsymbol b}_{222}}\; $$

Equation (8) presents the fundamental discovery of binodal astigmatism made by Shack [2]. The astigmatic aberration field contains two zeros, or nodes, in an optical system without symmetry. The nodes are located as illustrated in Fig. 1(a), and neither of the nodes (±ib₂₂₂) is necessarily located on the optical axis ray (OAR), which is an entirely new view of aberration fields in the optical system without symmetry. As seen in Fig. 1(a), the a₂₂₂ vector is directed from the optical axis ray to the midpoint of the ib₂₂₂ vectors in the image plane.

Fig. 1. (a) The characteristic field behavior of astigmatism in a misaligned optical system is for the nodes (points of zero astigmatisms, b₂₂₂) to be displaced in the image field around the point indicated by the vector a₂₂₂. (b) The coma-free pivot point of the secondary mirror brings one of the astigmatic nodes to the field center.

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Fig. 2. Schematic description of coma-free pivot (cfp) point for the secondary mirror of a two-mirror telescope with the aperture stop located on the primary.

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A unique astigmatic nodal property in misaligned Ritchey-Chrétien telescopes with misalignment coma removed was shown in NAT by Schmid et al. [14]. This special coma-removed state of the telescope can be used for isolating binodal astigmatism. There are many combinations of decentering and tilt of the secondary mirror that rotate the secondary mirror about a fixed point on the primary mirror’s optical axis that results in the elimination of coma, which is called the coma-free pivot point, as seen in Fig. 2. For the coma-free pivot point, one of the astigmatic nodes is placed onto the field center, while the second node is displaced in the image plane, as illustrated in Fig. 1(b). The a₂₂₂ vector is parallel to the ± ib₂₂₂ vectors, and one of the nodes falls back to the center for the coma-free pivot point.

The location of the secondary mirror coma-free pivot point, $L_{\textrm{SM}}^{({cfp} )}$, for a Cassegrain telescope system can be found utilizing the structural parameters of the telescope system as [15]

(9)$$L_{\textrm{SM}}^{({cfp} )} ={-} \frac{{YD{E_{SM}}}}{{AD{E_{SM}}}} = \frac{{{d_1}W_{131,SM}^{({sph} )}}}{{W_{131,SM}^{({asph} )} + {c_{SM}}{d_1}{W_{131,SM}}}}\; $$

where $L_{\textrm{SM}}^{({cfp} )}\textrm{is}\; $measured from the nominal vertex position of the secondary mirror, YDE_SM corresponds to decenter of the secondary mirror optical axis, ADE_SM corresponds to tilt angle of the secondary mirror optical axis, c_SM denotes the curvature of the secondary mirror, d₁ denotes the mirror spacing between the primary and secondary mirror, W_131,SM denotes the total coma contribution of the secondary mirror (i.e., W_131,SM = $W_{131,SM}^{({sph} )}$ + $W_{131,SM}^{({asph} )}$).

The field dependency of Fringe Zernike coma (Z_7/8) and astigmatism (Z_5/6) for a Cassegrain telescope system is illustrated in the form of full-field display (FFDs) for the aligned state in Figs. 3(a) and (b). Third-order coma is linearly dependent on the field, and third-order astigmatism shows quadratic dependency to the field as given in Eq. (1). The telescope gives a stigmatic image for the on-axis case, as seen from the figures. For the Cassegrain telescope's misaligned state, third-order coma remains symmetric and field-linear as in the aligned state, as shown in Fig. 3(c) if the tilt and decenter occur around the coma-free pivot point laid out in Eq. (9). The coma displays field dependency is equal to that of the aligned Cassegrain telescope. In this case, one of the astigmatic nodes is located on the field center, as seen in the FFDs of Fig. 3(d).

Fig. 3. Full Field Displays (FFDs) of Fringe Zernike coma Z_7/8 and astigmatism Z_5/6 in a Cassegrain system (a-b) in the aligned state, (c-d) in the misaligned state with a -0.43 mm decenter and 0.19° tilt for a coma-free pivot point that displays binodal astigmatism. The dots (blue) depict the nodes’ locations.

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3. Cassegrain telescope system

The telescope design is based on the Cassegrain optical layout, comprising a parabolic concave primary mirror and a hyperbolic convex secondary mirror. The realization of the Cassegrain telescope was discussed extensively in our previous work. The radii of the primary and the secondary mirrors are -1700 mm, and -300 mm, respectively. The conic constants are -1 and -1.737 for the primary secondary mirrors, respectively. The primary mirror defines the telescope system's aperture and is set as an aperture stop in the optical design. The telescope is diffraction-limited over ±0.11 deg. full field of view (FOV) and has an f-number of 12.7. The Cassegrain telescope system's specifications are given in Table 1, and the schematic optical layout of the system is provided in Fig. 4.

Fig. 4. Schematic layout of the Cassegrain telescope by design

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Table 1. Specifications of the Cassegrain telescope system

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An athermalized, high stability optomechanical structure was conceived for the Cassegrain telescope (see Fig. 7). It consists of a mainframe with support rods, a primary mirror assembly, metering rods, the secondary mirror ring, spider vanes, and a secondary mirror assembly. The primary mirror assembly comprises a lightweight primary mirror, glue pads, bipod flexures, and a base plate connected to the mainframe. The secondary mirror assembly comprises the secondary mirror, flexure stages, and a spider. The primary and the secondary mirror assemblies are connected by six metering rods, a secondary mirror ring, and spider vanes. The mainframe, secondary mirror ring, spider vanes, and spider are made from high-strength aluminum; the mirror supports and metering rods are made of low-thermal expansion Invar. The mirror material is Zerodur glass-ceramic.

Three planar surfaces were designed on the circumference of the primary mirror with 120◦ radial symmetry for the Invar glue pads that define the mechanical interface between the mounts and the mirror. The glue pads were bonded onto the mirror side surfaces utilizing 3M-2216 B/A gray epoxy adhesive before the surface finishing process. The support locations of the glue pads were aligned to pass through the neutral plane of the mirror in order to reduce the bending moments applied to the primary mirror. The bipod flexures, which were bolted to the glue pads, were designed to eliminate all figure errors originated by gravity, temperature, and mechanical stress. The surface finishing of both mirrors during the polishing process were completed for the assembled configuration of the telescope which eliminates all the induced stress on the mirrors. The final figure error of the mirrors was measured to be 0.013 and 0.006 wave RMS-WFE at 632.8 nm for the primary and secondary mirrors, respectively. The detailed descriptions of the system were given in our previous work [13].

The telescope system uses a customized, five-axis flexural alignment mechanism (X, Y, Z, and tip/tilt) to align the secondary mirror with respect to the primary mirror. The schematic design of the mechanism is illustrated in Fig. 5. The mechanism has two stages, and the first stage has two flexures that are designed to allow the secondary mirror for adjusting in the X and Y-axis with ±1 mm travel range. The second stage has three flexures designed to adjust tip/tilt/Z movements of the secondary mirror. Five piezo nanopositioners (PZA12 Actuators, Newport Corp., Irvine, CA) were integrated into these flexural for fine adjustments of the secondary mirror. The nominal mean step size of these nanopositioners was ∼30 nm. This fine alignment mechanism was also used to intentionally introduce the secondary mirror's misalignments to generate aberrations of misalignment in our study. Unlike many commercial hexapod systems, this piezo-based high-precision alignment mechanism does not dissipate much heat. Heat causes turbulence over the telescope system that is a nuisance.

Fig. 5. Schematic designs of (a) the piezo-actuated, five-axis flexural alignment mechanism for the secondary mirror (M2) and (b) the assembled alignment mechanism.

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4. Experimental setup and simulations

4.1 Experimental setup

An auto-collimation experimental setup was conceived, comprising the Cassegrain telescope, a reference flat mirror, and an interferometer. The optical layout of the setup is shown in Fig. 6. A phase-shifting Fizeau interferometer (DynaFiz, Zygo Corp., Middlefield, CT, USA) with a transmission sphere (F/7.1), which was confocal with the telescope, was placed on a custom-designed, five-axis manual stage (X, Y, Z, and tip/tilt) to move the interferometer focus to the image plane for the field tests. The telescope's output beam was reflected by a 32’’ reference flat mirror (QED Optics, Rochester, NY, USA) with a motorized tip/tilt stage and had < 0.015 wave RMS-WFE at 632.8 nm. The reference flat mirror was placed in the closest allowable proximity of the telescope to minimize the optical path and hence the turbulence effects. The entire system was placed on an optical table (3.6 m length and 1.5 m width) with a vibration isolation stage with a cut-off frequency of 1.5 Hz (S2000A, Newport Corp., Irvine, CA).

Fig. 6. Auto-collimation experimental setup in the double-pass interferometric measurement configuration and simulated field points on the image plane. The interferometer focus, which is always perpendicular to the image plane, moved on the field points, and the reference flat mirror was adjusted to be perpendicular to the output beam of the telescope for each field point utilizing the tip/tilt stage of the reference flat.

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The telescope alignment was performed utilizing a sensitivity matrix of the optical design and decreasing the Zernike terms of third-order aberrations for the on-axis position, which was discussed explicitly in our previous work [13]. The aligned telescope wavefront error was measured to be 0.03 wave RMS at 632.8 nm given in Fig. 7. The telescope and the experimental setup are shown in Fig. 8.

Fig. 7. The aligned wavefront error of the Cassegrain telescope (λ = 632.8 nm). Additional obscurations beside M2 were caused by the piezo nanopositioners of the flexure stages.

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Fig. 8. Custom Cassegrain telescope placed on the optical table in the auto-collimation setup configuration.

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The simulated field points of the 5 × 5 grid are given in Table 2. The field measurements were performed according to these simulated field points. For this purpose, a custom-designed apparatus was used, which was mounted onto the rear plane of the primary mirror, to guide the focus of the interferometer. The end surface of the guide apparatus was perpendicular to the optical axis and placed at the image plane of the telescope. The field points were depicted on the image plane (see Fig. 6). The focus of the interferometer beam was moved across the image plane utilizing the five-axis manual stage beneath the interferometer. Four points located at the corners were excluded from the simulations and experiments because of the primary mirror's obscuration. The inner hole dimension was not enough to get the full beam for the interferometric measurements since the simulation fields were beyond the design FOV limit of the telescope.

Table 2. 5 × 5 grid of field positions on the image plane simulated.

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4.2 Simulations

A real ray trace model of the Cassegrain telescope and the Wavefront Map Analysis in Zemax were used to get the Zernike Fringe Coefficients. In order to orient the results with the experimental interferometric outcomes, ‘Zernike Fringe Coefficients’ were used instead of ‘Zernike Standard Coefficients’ for the 5 × 5 grid of field points given in Table 2. The simulation results were plotted separately for coma and astigmatism. The 5^th and 6^th Fringe Zernike terms were used to quantify astigmatism as given in Eq. (10), and the 7^th and 8^th Fringe Zernike terms were used to quantify the magnitude of coma as given in Eq. (11). The interferograms shown in Figs. 9(a) and (b) were obtained by a customized Matlab code collecting the individual results into a 5 × 5 grid. The interferograms show the comma and astigmatism's magnitude and orientation for the 5 × 5 grid of field points in the misaligned state. The misalignment values of -0.43 mm decenter in XDE, and 0.19° tilt in ADE were introduced to obtain binodal astigmatism for a coma-free pivot point. For this point, as seen in Fig. 9(a), the on-axis coma remains zero (as in the aligned state), and the coma shows a linear dependency with the increasing field, which is expected for a Cassegrain telescope. As seen in Fig. 9(b), one of the astigmatic nodes was located on the field center, and the second node was located at the edge of the image plane for the coma-free pivot point. We maximized the amounts of misalignment up to the point where measuring the two astigmatic nodes’ midpoint could no longer be executed. The limit for this level of misalignment was the inner diameter dimension of the primary mirror.

(10)$$|{{Z_{5/6}}} |= \sqrt {Z_5^2 + Z_6^2} \; $$

(11)$$|{{Z_{7/8}}} |= \sqrt {Z_7^2 + Z_8^2} \; $$

Fig. 9. Simulated interferograms of (a) Fringe Zernike coma (Z_7/8) and (b) astigmatism (Z_5/6) in a misaligned state of the Cassegrain telescope (λ = 632.8 nm).

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5. Experimental results

The interferometric measurements were performed according to simulated field points given in Table 2, and the corresponding interferograms were obtained and displayed in Fig. 10. Utilizing the measurement data, the coma and astigmatism were calculated and presented in Figs. 11(a) and (b). The 5 × 5 interferograms in the figure were obtained with the same method as the analysis results. The Zernike coefficients obtained from the interferometer, which are also Zernike Fringe coefficients [16], were plotted separately and combined with the Matlab code. The experimental results are consistent with the simulations. Figure 11(a) validates that coma is zero on-axis and follows the field-linear coma for the misaligned state of the Cassegrain telescope, as simulated. In Fig. 11(b), binodal astigmatism is presented over the image plane. One of the astigmatic nodes was found on the field center, and the second node was displaced at the edge of the image plane, as also predicted in simulation. The midpoint of the nodes could be clearly measured for the introduced misalignments. There are negligeable variations in the order of one to two 100^th of a wave between the experimental results and simulations as seen in Fig. 12, originated mainly by the interferometer positioning error focus on the predetermined simulation field points using the manual stage. The other error source is generated by introducing the secondary mirror misalignments utilizing the secondary mirror alignment mechanism to employ the simulated coma-free pivot point. While performing the experiments, we observed that even a small change in misalignment values changes both the amplitudes and orientation for astigmatism.

Fig. 10. Experimental results of interferograms in a misaligned state of the Cassegrain telescope for the coma-free pivot point of the secondary mirror. The values on each figure show the RMS WFE of the telescope for the field points.

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Fig. 11. Processed measured interferograms of (a) Fringe Zernike coma (Z_7/8) and (b) astigmatism (Z_5/6) in a misaligned state of the Cassegrain telescope for the secondary mirror. The secondary mirror was decentered -0.43 mm in XDE and tilted 0.19° in ADE (λ = 632.8 nm).

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Fig. 12. Simulation and experimental full field display (FFD) matching of binodal astigmatism (Z_5/6) for the coma-free pivot point. The blue dots show the node positions.

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The location of each astigmatism node in the image plane was calculated analytically according to Eqs. (2)–(6) as given in Table 3. The results are compared with the node locations calculated from the experimental data given in Fig. 10, and they are within three hundredths of a degree. Hence, experimentally validating NAT for binodal astigmatism.

Table 3. Location of astigmatism nodes as predicted analytically and experimentally.

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Thompson et al. [17] introduced the aberration field-asymmetric, field-linear, 3^rd order astigmatism in NAT for astigmatism-corrected three-mirror anastigmat (TMA) telescopes. This new, field-depended 3^rd order astigmatism can be validated indirectly here utilizing the data collected. The difference between the ideal telescope and the measured astigmatism in the misaligned Cassegrain gives the field-asymmetric, field-linear 3^rd order astigmatism. For this purpose, the 3^rd order astigmatism in the misaligned telescope seen in Fig. 13(b) was subtracted from the ideal telescope simulation data given in Fig. 13(a), and the field-asymmetric, field-linear 3^rd order astigmatism seen in Fig. 13(c) was obtained.

Fig. 13. (a) Fringe Zernike astigmatism (Z_5/6) in the aligned state of the Cassegrain telescope from the simulation, (b) processed measured interferograms of Fringe astigmatism (Z_5/6) in a misaligned state for the coma-free pivot point of the secondary mirror, and (c) the difference between astigmatism in aligned state and misaligned state which shows field-asymmetric, field-linear astigmatism.

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Statistical analysis of the measurements was also performed to see the influence of the environment on the stability of the results. Influences such as air turbulence, vibration, and temperature were considered. The measurement results were analyzed to verify their accuracy and stability. For this purpose, multiple measurements (i.e., ten consecutive measurements) were performed at the field point (i.e., 0°, -0.25° field point) in a misaligned state. As seen in Fig. 14(a), a line chart of the three parameters, Fringe Zernike astigmatism pair Z_5/6, Fringe Zernike coma pair Z_7/8, and the RMS wavefront error are given for ten consecutive measurements. Figure 14(b) reports the deviations from the mean values for these three variables, which indicates the standard deviations for Z_5/6, Z_7/8, and RMS-WFE are 0.003, 0.001, and 0.002 waves at 632.8 nm, respectively.

Fig. 14. Error analysis: (a) Statistical analysis of the measurements at field point (0°, -0.25°) in the misaligned state of the Cassegrain telescope, (b) the deviations from the mean values for ten consecutive measurements for that field point in three different variables (λ = 632.8 nm).

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When the Zernike Fringe fit error analysis is considered for the annular aperture case in which the obscuration is not zero, we calculate an insignificant contribution to the fitting error. For our Cassegrain telescope, the secondary mirror obscuration is 20%, and the overall obscuration, including the secondary mirror’s spider contribution, was calculated to be 27%. This ratio was similar to the case study by Hou et al. [18], and the fits are similar for both circular and annular Zernike fits.

6. Conclusions

In this work, we presented the first experimental investigation of binodal astigmatism in NAT for a customized, high-precision Cassegrain telescope. The characteristics of binodal astigmatism were studied with an intentionally misaligned state of the telescope. The misalignment-induced aberrations were simulated and measured at twenty-five field points to validate the third-order astigmatism's binodal characteristic. The location of each astigmatism node was calculated both analytically and from experimental data, which are consistent with each other within three hundredths of a degree. Simultaneously, linear field dependency of third-order coma was investigated experimentally for the coma-free pivot point of the misaligned Cassegrain telescope. In addition, the new field-depended aberration called the field-asymmetric, field-linear 3^rd order astigmatism for TMA telescopes was validated indirectly utilizing the data collected on the Cassegrain telescope. The experimental results were also analyzed statistically to show the accuracy and stability of the measurements. Per the measurements and the analysis, it was found that the environment was not the limiting factor of uncertainty, rather manual adjustments during the experiment were dominant, yet in the 100^th of a wave magnitude. As NAT predicts, in a two-mirror misaligned telescope where the secondary moves around the coma-free pivot point, astigmatism displays binodal behavior for the fields in the image plane with one node located at the origin.

Funding

Fulbright Association (FY-2019-TR-PD-06).

Acknowledgments

Özgür Karcı thanks the Fulbright Commission for their partial support and the University of Rochester for their hosting during his Fulbright fellowship. Özgür Karcı also thanks Kevin P. Thompson, Jonathan C. Papa, Nan Zhao, Yuchen Wu, and Jannick P. Rolland for their great contributions to the RC telescope setup in which all the inaugural studies were later conducted on the Cassegrain telescope.

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

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Parameter	Magnitude
Aperture (mm)	490
Wavelength (nm)	632.8
Full field of view (deg)	± 0.11
Secondary obscuration (linear diameter)	20%
Effective focal length (mm)	6200
Overall length (mm)	944

(-0.25°, 0.25°)	(-0.125°, 0.25°)	(0°, 0.25°)	(0.125°, 0.25°)	(0.25°, 0.25°)
(-0.25°, 0.125°)	(-0.125°, 0.125°)	(0°, 0.125°)	(0.125°, 0.125°)	(0.25°, 0.125°)
(-0.25°, 0°)	(-0.125°, 0°)	(0°, 0°)	(0.125°, 0°)	(0.25°, 0°)
(-0.25°, -0.125°)	(-0.125°, -0.125°)	(0°, -0.125°)	(0.125°, -0.125°)	(0.25°, -0.125°)
(-0.25°, -0.25°)	(-0.125°, -0.25°)	(0°, -0.25°)	(0.125°, -0.25°)	(0.25°, -0.25°)

Parameter	X-component	Y-component
Analytical Node 1 (degree)	-0.0001	0
Experimental Node 1 (degree)	0.0102	0
Analytical Node 2 (degree)	-0.2342	0
Experimental Node 2 (degree)	-0.2613	0.0086

Parameter	Magnitude
Aperture (mm)	490
Wavelength (nm)	632.8
Full field of view (deg)	± 0.11
Secondary obscuration (linear diameter)	20%
Effective focal length (mm)	6200
Overall length (mm)	944

(-0.25°, 0.25°)	(-0.125°, 0.25°)	(0°, 0.25°)	(0.125°, 0.25°)	(0.25°, 0.25°)
(-0.25°, 0.125°)	(-0.125°, 0.125°)	(0°, 0.125°)	(0.125°, 0.125°)	(0.25°, 0.125°)
(-0.25°, 0°)	(-0.125°, 0°)	(0°, 0°)	(0.125°, 0°)	(0.25°, 0°)
(-0.25°, -0.125°)	(-0.125°, -0.125°)	(0°, -0.125°)	(0.125°, -0.125°)	(0.25°, -0.125°)
(-0.25°, -0.25°)	(-0.125°, -0.25°)	(0°, -0.25°)	(0.125°, -0.25°)	(0.25°, -0.25°)

Experimental investigation of binodal astigmatism in nodal aberration theory (NAT) with a Cassegrain telescope system

Abstract

1. Introduction

2. Nodal property of third-order astigmatism in a misaligned optical system

3. Cassegrain telescope system

4. Experimental setup and simulations

4.1 Experimental setup

4.2 Simulations

5. Experimental results

6. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (14)

Tables (3)

Equations (11)

Optics Express