Theory and design of Schwarzschild scan objective for Optical Coherence Tomography

: Optical coherence elastography (OCE) is one form of multi-channel imaging that combines high-resolution optical coherence tomography (OCT) imaging with mechanical tissue stimulation. This combination of structural and functional imaging can require additional space to integrate imaging capabilities with additional functional elements (e.g., optical, mechanical, or acoustic modulators) either at or near the imaging axis. We address this challenge by designing a novel scan lens based on a modified Schwarzchild objective lens, comprised of a pair of concentric mirrors with potential space to incorporate additional functional elements and minimal compromise to the available scan field. This scan objective design allows perpendicular tissue-excitation and response recording. The optimized scan lens design results in a working distance that is extended to ~140 mm (nearly 2x the focal length), an expanded central space suitable for additional functional elements (>15 mm in diameter) and diffraction-limited lateral resolution (19.33 μ m) across a full annular scan field ~ ± 7.5 mm to ± 12.7 mm.


Introduction
A Multi-channel optical system contains several independent working channels with various functionalities, such as illumination, mechanical stimulation, imaging, etc. In biomedical and clinical studies, the independent imaging channels could include modalities, such as confocal microscopy, ultrasound, x-ray, magnetic resonance imaging, fluorescence intensity imaging, two-photon imaging, and optical coherence tomography. Compared to a specific singlechannel imaging system, multi-channel optical systems can provide complementary, synergistic information, or enable rapid switching between different modes and functions [1]. The use of a multi-channel optical system is helpful to expand the potential uses of noninvasive imaging. For example, dynamic elasticity imaging systems are used to determine tissue mechanical properties (e.g., stiffness) [2,3] by combining a mechanical loading channel (a source of sample stimulation) and an imaging channel to record the sample response.
Optical coherence elastography (OCE) [4] is an emerging elasticity imaging technique that employs at least two channels. A loading channel is used to induce elastic waves in a tissue using techniques, such as optical, mechanical, or acoustic modulators for sample stimulation. Various approaches of loading methods have been developed, such as static [5,6], dynamic contact [4,[7][8][9], audio sound [10], pulsed laser [11,12], and air puff/pulse [13][14][15] loading. The second channel uses optical coherence tomography (OCT) [16] imaging to record the tissue response. Compared to traditional ultrasound elastography [17][18][19] and magnetic resonance elastography [20,21], OCT can noninvasively obtain tissue mechanical properties with higher spatial resolution and faster speeds [22]. Phase-sensitive OCT methods [23][24][25][26][27] have further improved the dynamic surface displacement detection sensitivity to a sub-nanometer scale. For example, we reported a 0.24 nm resolution in our common-path OCE results [28]. More channels may be added into the OCE system for specific purposes. For example, a targeting channel and a monitoring camera can be used for locating the regions of stimulus and imaging in the tissue.
The combination of a loading mechanism and OCT imaging usually requires space between the OCT scan objective and sample, especially for dynamic/transient, non-contact OCE applications [10][11][12][13][14][15]. An optimal OCE set-up should satisfy certain criteria. First, the loading channel should be set up to deform tissue in a predictable way so that tissue mechanical properties can be derived from the deformation response. Loading normal to the surface is advantageous since it simplifies the complex modeling methods that are used to derive the mechanical properties from the observed response [4][5][6]. Second, measurement distance to the stimulation point should be optimized to clearly record the induced wavepropagation and to avoid near-field effects [29]. Capability of measuring around the stimulation point would also be advantageous to determine tissue anisotropy.
However, spatial conflict often occurs between the physical bulk of the loading system and a limited space provided by the OCT system. Consequently, oblique tissue-excitation has been adopted by several investigators instead of the preferred perpendicular tissue-excitation [7,8,[11][12][13][14]. The axial distance between the OCT scan objective and sample is usually similar to, or shorter than, the focal length of the scan objective lens. The focal length is often relatively short to achieve a desired optical lateral resolution. For instance, the focal length of the OCT scan lens of our OCE system in [28] was 54 mm, and the total working distance was 42 mm. Therefore, designing a scan objective with a longer working distance without sacrificing optical performance is important for OCE imaging to quantify tissue biomechanics.
Here we describe a novel OCE scan objective comprised of a pair of concentric convex and concave mirrors. This reflective objective is a modification of a Schwarzschild lens design [30]. First-order theory is used to determine the general geometric parameters, especially the focal length, working distance, and dimensional constraint criteria of the Schwarzschild scan objective. Astigmatism for the marginal rays of each scan beam is derived and minimized based on the Coddington equations [31,32]. Optical path differences (OPDs) among all scan beams are reduced to provide effective OCT interference signals from reference and sample arms. The Schwarzschild scan objective extends the working distance and enables adequate free space to accommodate a loading system that can deliver force normal to the tissue surface. Since all the optical elements are mirrors, this Schwarzschild scan objective is free of chromatic aberration and is idea for applications in systems with broad bandwidth (e.g. from visible to near-infrared range that is usually applied in OCT systems).
A Schwarzschild scan objective is designed for, but not limited to, OCE imaging systems. It may also benefit other multi-channel imaging systems that combine peripheral scans with central channels of various purposes. For example, a camera situated in the center area, enclosed by the peripheral scan beams, can serve as a view-finder to guide the scan beams to specific locations. An illumination light source situated in the center of the scan beams may also benefit fluorescence or two-photon imaging.

Schwarzschild scan objective
The Schwarzschild system [30] was initially designed for astronomical telescopes and was more recently adopted for use in microscope objectives [33][34][35][36].The Schwarzschild system consists of two mirrors, as demonstrated in Fig. 1. Previous publications have discussed the use of the Schwarzschild design to correct Seidel aberrations, such as spherical aberration, coma, astigmatism and distortion, and to provide a flat field when the object is either in an infinite or finite distance [33][34][35][36].  There are some additional concerns in the design of the Schwarzschild scan objective compared to the classic Schwarzschild construction.
(1) The scan angles (θ min to θ max ) and the scan zone at the focal plane ( ± H min to ± H max ) are constrained by the physical size and distance of the two mirrors. For example, the size of Mirror 1 should be big enough to reflect the beam with the maximum scan angle (θ max ) and should be also small enough to ensure the passage of the beam with the minimum scan angle (θ min ) in a non-vignetting condition. Therefore, the modified design incorporated these additional dimensional constraints and established a new criterion to meet the spatial requirements for the Schwarzschild scan objective design.
(2) The light path of the chief ray for each scan beam in the newly-designed Schwarzschild scan objective ( Fig. 2(a)) is similar to the light path of each ray in the classic paraxial Schwarzschild construction (Fig. 1). We used methods described in previous work [33][34][35][36] to correct Seidel aberrations of the chief rays. However, the layout for each scan beam ( Fig. 2(b)) is off-axis with incident angles which can be tens of degrees to the normal of each mirror. Also, the incident angles for all of the scan beams are also over a wide range (tens of degrees). In this application, astigmatism becomes the major contribution to the total aberrations [37] of each scan beam, and the value of astigmatism varies by scan location. Therefore, minimization of astigmatism across the entire scan range was a major design goal.
(3) This Schwarzschild scan objective was developed for an OCT-based elastography application. OCT imaging is a form of low-coherence interferometry where the interference signal is generated by combining light from reference and sample arms [38]. Effective interference requires minimal optical path difference between these two arms. Therefore, the light path length difference among the chief-rays of all the scan beams was constrained to meet the requirement of OCT detection.

Key points for the Schwarzschild scan objective design
The design principles for the Schwarzschild scan objective are provided as follows: (1) First-order optical design principles were used to determine the general geometry of the Schwarzschild scan objective that would maximize the axial working distance (defined by the radii of the mirrors (Fig. 2)) without greatly enlarging the radial dimensions of the scan lens for a specified scan range. The extended working distance was derived by calculating the axial dimensions and the radial dimensions were primarily determined by the defined scan range.
(2) The Coddington equations [31,32] were applied to quantify the astigmatism, defined as the difference in focal distance between the tangential and sagittal marginal rays [39,40] for each scan beam. The astigmatism value across the whole scan range was then evaluated and minimized by optimizing the design parameters.
(3) The chief-ray optical length (OPL) was calculated for each scan beam. The optical path differences (OPDs) of all the chief-rays from all of the scan beams were estimated. The OPD values were balanced with other design requirements (e.g. size of the system, astigmatism) to enable the use of the Schwarzschild scan objective in OCT imaging. The detailed theoretical analysis for the above design concerns are discussed in the following sections.

Axial dimensions and working distance extension
The radii of curvatures for Mirror 1 and Mirror 2 are R 1 and R 2 , respectively, both are defined as positive values. The focal lengths of these two mirrors are f 1 = -R 1 / 2 and f 2 = R 2 / 2. We define R 2 = M × R 1 (M >1), where M is ratio of the raddi of curvatures as well as the ratio of focal lengths between two mirrors. The distance between scanner and Mirror 1 is d 1 . The distance between Mirror 2 and Mirror 1 is d 2 , where d 2 = R 2 -R 1 because Mirror 1 and Mirror 2 are concentric. The total focal length f of the objective can be calculated as: . The exit pupil distance d exp is defined as the axial distance from Mirror 1 to the exit pupil, and the working distance d work is defined as the axial distance from Mirror 1 to the focal plane. d exp and d work can be expressed as: where T 1 and T 2 are the center thicknesses for the two mirrors. Comparing Eq. (1) and Eq. (3), we have: 1 1 , If R 1 > T 1 , then d work > f. Therefore, the working distance can be extended to a longer value than the focal length. For the same value of f, d work is increased by the same amount as R 1 is increased. Therefore, f is equal to the distance from the mutual center point of the two mirrors to the focal plane. The total length L from Mirror 2 to the focal plane can be expressed as:

Radial dimensions and dimensional constraint criteria
Scan angle θ is in the range of θ min to θ max , its corresponding scan length H (H = f tanθ) at the focal plane is in the range of H min to H max . Figure 2 depicts RH 1 to RH 5 , the specific marginal ray heights. RH 1 is the maximum ray height inside the center hole of Mirror 2. RH 2 and RH 3 are the inner and outer ray heights for the light annulus on Mirror 2. RH 4 is the maximum ray height at the front surface of Mirror 1, and RH 5 is the minimum ray height at the back surface of Mirror 1. These marginal ray heights can be expressed as: where D 0 is the entrance pupil size or beam size, D 1 is the diameter of Mirror 1, and D 2_out and D 2_in are the outer and inner diameters for Mirror 2. The expressions for RH 1 to RH 5 all consist of two components. The first component contains either tanθ max or tanθ min , and denotes the chief ray heights. The second component contains D 0 , and denotes the half beam size at the corresponding surface.
To avoid vignetting in Mirror 1 and Mirror 2, the following dimensional criteria should be satisfied: where ρ 1 , ρ 2 , and ρ 2_out are the ratios for the clear aperture (usually, ρ ≈90%). The first criterion defines the minimum requirement for D 2_out . The second criterion ensures the vignetting-free condition for Mirror 2 where the scan beams can go through the center hole, and the reflected beams from Mirror 1 can reach the effective optical portion of Mirror 2. The third criterion determines the vignetting-free condition for Mirror 1 so that the scan beams can be reflected by the clear aperture of Mirror 1, while the reflected light from Mirror 2 can bypass the outside diameter of Mirror 1.

Optical path length (OPL) and optical path difference (OPD)
To guarantee the effective interference between the signals from the sample arm (where OPL varies across the field of view) and the reference arm (where OPL is a constant value), the OPD from all the chief-rays across the scan field in the sample arm must be limited to a certain range. As shown in Fig. 2, The OPL for this chief ray is dominated by R 1 and θ and can be expressed as: The maximum OPD from all the chief-rays across the scan field can be expressed as:

Astigmatism
As shown in Fig. 2, the layout for each scan beam is off-axis with large incident angles relative to the surface normal for each mirror. In this mirror-based, off-axis construction, the dominant aberration is astigmatism [37], defined as the difference in focal distance between the tangential and sagittal marginal rays [39,40]. Applying the Coddington equations [31,32] to Mirror 1 and Mirror 2, we can express the general astigmatism (AST) as: Corresponding to the different scan angle θ, the value of astigmatism varies across the scan field. We define a discrete mean absolute astigmatism (DMAA) equation to minimize the total amount of astigmatism in the required scan range: where θ i = θ min + i(θ max -θ min )/k, k is the step number (i = 0, …, k), and α i is the weight.

Quantitative simulation
We performed a quantitative simulation to demonstrate the design of the Schwarzschild scan objective. We first specified the design requirements and then computed the possible R 1 values that met each of the design requirements separately. We then optimized R 1 to simultaneously satisfy all of the requirements.

Design requirements
The design parameters for this Schwarzschild scan objective were defined by the current specifications of our clinical OCE system [28]. The spectral bandwidth of the OCT light source was 795 nm -895 nm, D 0 = 4 mm, and f = 75 mm, the lateral resolution was 19.33 μm, calculated at the central wavelength of 845 nm. The requirements for the loading and scanning space were specified as d work ≥ 120 mm, 2H min ≥ 10 mm, and H max to H min ≈5 mm.
The dimensional requirements were specified as D 1 ≤ 70 mm, D 2_out ≤ 200 mm, T 1 = 10 mm, T 2 = 20 mm -30 mm, and L ≤ 300 mm. The distance requirements for the X and Y scanners were set to d 1 keep the stru objective was between the s the entire FOV

Computi
The key varia Here, we redu dimensions an      of the value of was decreased within the R 1 ra n was stricter atisfaction of th set to a valu quirement of th θ min to θ max ) an en R 1 was incr th Fig. 4

Calculation summary and optimization of R 1
The requirements for the key design parameter R 1 were: (1) To achieve the dimensional requirements of L ≤ 300 mm and d work ≥ 120 mm, R 1 was limited to the range of 55 mm to 88.36 mm.
(2) To relate all of the scan-related parameters (θ min to θ max , H min to H max , D 1 , D 2_out , OPD, and astigmatism) to R 1 , the minimum requirement of the third criterion in Eq. (8) was used, where 2RH 4 /ρ 1 = 2RH 5 .
(3) To meet the diameter requirements of D 1 ≤ 70 mm and D 2_out ≤ 200 mm, R 1 was required to be less than 79.60 mm.
(5) To reduce the astigmatism for the entire scan field, R 1 was required to be in the range of 80 mm to 90 mm. To satisfy requirements (1) to (5), R 1 was found to be optimal in the range 55 mm to 79.60 mm, where the requirements for L, d work , D 2_out and OPD max were met. However, this range of values for R 1 did not minimize astigmatism. Since the residual astigmatism decreased as the value of R 1 was increased in the range 55 mm -79.60 mm (Fig. 5 (f)), we chose a relatively large R 1 value (75 mm). Note that, aspherical surfaces can be used to further minimize residual astigmatism and other aberrations.

Zemax validation and design results
We employed the optical design software Zemax (Zemax, LLC) to validate the calculation, and to finalize the design. Mirror 1 was designed as an aspherical mirror to further reduce aberrations. A low-order standard aspherical surface is given by [41]: where the optical axis is presumed in the z direction, z(r) is the sag value (the displacement of surface in the z direction from the vertex at a distance of r from the optical axis), and r is in the range 0 to 0.5ρ 1 D 1 . The conic constant k 1 of the aspherical Mirror 1 was set as the only variable to be optimized, along with all of the other parameters that were calculated in Section 3. The aspherical surface type is determined by k 1 , and can be hyperbola (k 1 < -1), parabola (k 1 = -1), prolate ellipse (-1 < k 1 < 0), sphere (k 1 = 0), or oblate ellipse (k 1 > 0). Using the two scanners, fields in the x respectively. scan field in b k 1 > 0), as sho All of the ensuring no c collimated by Galvo mirror through a 1-in radius, and th for Mirror 1 apertures wer The thickness mounted usin to the focal pl    Fig. 9. Demonstration of the tissue-excitation (loading) and the wave-detection (scanning) areas, as well as the spot diagrams at the focal plane, simulated in Zemax. The purple stars show the possible loading locations. The distance between two spots is 1mm in the x and y directions. The shadow areas are due to the obscuration of the Mirror 1 mount.

Non-uniform illumination due to angle of incidence
We used a telecentric scan lens in our previous OCE system [28] where the chief rays of scan beams were parallel to the optical axis and perpendicular to the focal plane. Compared to the non-telecentric construction, the telecentric scan lens can illuminate the sample more uniformly and collect more reflected/scattered light back to the system with a flat sample geometry (e.g. 2% agar phantom [28]).
In the Schwarzschild scan objective, scan beams are not perpendicular to the focal plane and the incident angel (θ') varies across different scan positions. In the design example of Section 4, the incident angels at the focal plane vary from 9.88° to 19.14° in the x direction, and from 10.09° to 19.05° in the y direction. Illumination is positively correlated with cos 4 (θ'). When the sample surface was flat, the illuminance values were calculated as 94.2% -79.7% in the x direction, and 94.0% -79.8% in the y direction, relative to the perpendicular illumination.
Tissue samples may have any shape including convex surface geometry, such as the cornea [42]. In our previous work, we have quantified the biomechanical properties of rabbit [43][44][45] and porcine [46][47][48][49] corneas. We noticed that the imaging intensities and phase sensitivities dropped noticeably when imaging away from the apex or in the peripheral regions of the cornea using a telecentric scan lens. In this case, a scan lens with a convergent scan beam geometry would provide greater tissue illumination and back-light collection for better image contrast than a telecentric scan lens. The design example presented in section 4 has a larger work space relief to accommodate a loading channel for corneal OCE applications. Future designs could also include a convergent beam scan lens for other applications.

Design alternative I: optimizing d 1 to further reduce OPD
In Section 3.3, the required R 1 range was found to be 55 mm to 79.60 mm. Instead of choosing a smaller value (for example 55 mm) to reduce the OPD, we selected a value of 75 mm to reduce astigmatism. This resulted in a maximum absolute OPD of 1.34 mm, across the entire scan field (Fig. 8 (b)).
Without compromising the correction of astigmatism, another possibility for OPD reduction would be to adjust the value of d 1 . Our calculations and simulations demonstrated that a smaller d 1 reduced OPD. However, the scanning mirrors in our design were big and had to be located to the left side of Mirror 2 (Fig. 7), resulting in a relatively big d 1 (120 mm and 134.7 mm, respectively for the two scanners). Using smaller scanners and mounts might reduce d 1 and further reduce OPD, if they can be located between Mirror 1 and Mirror 2.

Design alternative II: using spherical surfaces for both mirrors
For astigmatism minimization, we designed Mirror 1 as a standard aspherical mirror with a small diameter (56 mm) and small departures (k = 0.436) from a standard spherical mirror. The manufacturing cost for such a mirror is inexpensive.
If a larger R 1 value is chosen, such as 90 mm, both mirrors can be spherical surfaces and still achieve diffraction-limited performance. However, a larger R 1 will require a larger diameter Mirror 2 (> 300 mm). This would greatly increase the cost of manufacturing and mirror verification.

Design alternative III: splitting Mirror 2 as multiple small mirrors
Some specific applications may require better lateral resolution (related to D 0 /f) or a need to accommodate larger scan areas (related to scan angles). Consequently, Mirror 2 with a larger aperture (e.g. > 200 mm) may be required to meet these demands. However, fabrication of a larger aperture of Mirror 2 is more difficult and expensive. A possible solution is to split Mirror 2 into multiple small mirrors, e.g. two or four mirrors. This mirror-splitting method can reduce the fabrication cost, but will require more complex construction as well as higher assembly cost.

Design alternative IV: mounting Mirror 1 onto glass to reduce obscuration
Holder vanes are commonly applied for mounting the small mirror into a two-reflector telescope, such as in the construction of Schwarzschild [30] and Cassegrain [50] lenses. Since the small mirror is usually located at the pupil plane in a conventional telescope, holder vanes do not generate blind spots at the image plane.
In our design, since Mirror 1 is not located at the pupil plane (Fig. 7), the three-arm holder vanes would induce obscurations at the focal plane (Fig. 9). Use of a flat glass window to mount Mirror 1 can effectively avoid such obscurations. We have performed the Zemax simulation using a 10-mm thick glass window (material: BK7) to mount Mirror 1 instead (wavelength: 795-895 nm). The point spread functions in the same scan areas are still diffraction-limited with only slightly increased chromatic dispersions over this wavelength range and this validates this glass window mount as a viable option. In addition, the reflective collimator used in Fig. 7 can be replaced by an achromatic lens-based collimator for this wavelength range as well. Choosing the mounting method would be determined by the specific design requirements, such as wave bandwidth, numerical number, scan angles, and complexity, as well as the preference of the designer.

Conclusion
We demonstrated the theory and design for an OCE reflective scan objective by employing a Schwarzschild design with two concentric convex and concave mirrors. This Schwarzschild scan objective extended the working distance, and enabled the use of central perpendicular tissue-excitation with peripheral wave-detection.
We presented a detailed theory in Section 2, where R 1 was chosen as the key value to calculate, optimize, and evaluate the main parameters, such as the axial dimensions, radial dimensions, radial constraints and dimensional criteria, OPD among chief rays, and astigmatism values for the scan beams.
In Section 3, the relation between the axial dimensions (L and d work ) and R 1 was demonstrated. The equation of 2RH 4 /ρ 1 = 2RH 5 was used to further relate R 1 to other parameters, such as scan angles, scan lengths at the focal plane, OPD among chief rays, and astigmatism for the scan beams. After balancing all of these key design constraints, especially D 2_out , we chose R 1 = 75 mm.
In Section 4, we defined the conic constant for Mirror 1 surface as the only variable, and optimized this scan objective to further reduce astigmatism and the residual aberrations (Fig.  6). The reflective configuration provided a chromatic-aberration-free design. This feature enables its use in broad bandwidths (e.g. from visible to near-infrared range that is usually applied in OCT imaging). Lateral distortion (relation between the scan angle and scan length) and axial distortion (OPD) were presented for system calibration (Fig. 8). Regions of loading and scanning were defined and the diffraction-limited performance was achieved (Fig. 9).
In summary, a Schwarzschild scan objective was designed for broad bandwidth optical coherence elastography with a long working distance, central perpendicular tissue-excitation, and peripheral wave-detection. OPD of the chief rays and astigmatism values for the scan beams were reduced and the lateral resolution was diffraction-limited for the entire scan field. This Schwarzschild scan objective may also benefit other multi-channel imaging systems that combine peripheral scans with other central channels.

Funding
This study was supported by the following grants: NIH/NEI R01-EY022362, P30EY07551, and P30EY003039 from United States; and the Foshan University start-up fund Gg07071 from China