Theory and design of Schwarzschild scan objective for Optical Coherence Tomography

Gongpu Lan; Michael D. Twa

doi:10.1364/OE.27.005048

Theory and design of Schwarzschild scan objective for Optical Coherence Tomography

Gongpu Lan and Michael D. Twa

Optics Express
Vol. 27,
Issue 4,
pp. 5048-5064
(2019)
•https://doi.org/10.1364/OE.27.005048

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Gongpu Lan and Michael D. Twa, "Theory and design of Schwarzschild scan objective for Optical Coherence Tomography," Opt. Express 27, 5048-5064 (2019)

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Related Topics
Table of Contents Category
- Optical Design and Fabrication
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: November 23, 2018
- Revised Manuscript: January 24, 2019
- Manuscript Accepted: January 26, 2019
- Published: February 11, 2019

Accessible

Open Access

Abstract

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 μm) across a full annular scan field ~ ± 7.5 mm to ± 12.7 mm.

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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
B. I. Akca, E. W. Chang, S. Kling, A. Ramier, G. Scarcelli, S. Marcos, and S. H. Yun, “Observation of sound-induced corneal vibrational modes by optical coherence tomography,” Biomed. Opt. Express 6(9), 3313–3319 (2015).
[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
S. Wang, K. V. Larin, J. Li, S. Vantipalli, R. K. Manapuram, S. Aglyamov, S. Emelianov, and M. D. Twa, “A focused air-pulse system for optical-coherence-tomography-based measurements of tissue elasticity,” Laser Phys. Lett. 10(7), 075605 (2013).
[Crossref] [PubMed]
S. Wang, J. Li, R. K. Manapuram, F. M. Menodiado, D. R. Ingram, M. D. Twa, A. J. Lazar, D. C. Lev, R. E. Pollock, and K. V. Larin, “Noncontact measurement of elasticity for the detection of soft-tissue tumors using phase-sensitive optical coherence tomography combined with a focused air-puff system,” Opt. Lett. 37(24), 5184–5186 (2012).
[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
K. Nightingale, S. McAleavey, and G. Trahey, “Shear-wave generation using acoustic radiation force: in vivo and ex vivo results,” Ultrasound Med. Biol. 29(12), 1715–1723 (2003).
[Crossref] [PubMed]
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[Crossref] [PubMed]
M. R. Ford, W. J. Dupps, A. M. Rollins, A. S. Roy, and Z. Hu, “Method for optical coherence elastography of the cornea,” J. Biomed. Opt. 16(1), 016005 (2011).
[Crossref] [PubMed]
Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. de Boer, and J. S. Nelson, “Phase-resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25(2), 114–116 (2000).
[Crossref] [PubMed]
S. J. Kirkpatrick, R. K. Wang, and D. D. Duncan, “OCT-based elastography for large and small deformations,” Opt. Express 14(24), 11585–11597 (2006).
[Crossref] [PubMed]
R. K. Wang, Z. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[Crossref]
R. K. Wang and A. L. Nuttall, “Phase-sensitive optical coherence tomography imaging of the tissue motion within the organ of Corti at a subnanometer scale: a preliminary study,” J. Biomed. Opt. 15(5), 056005 (2010).
[Crossref] [PubMed]
M. H. D. Torre-Ibarra, P. D. Ruiz, and J. M. Huntley, “Double-shot depth-resolved displacement field measurement using phase-contrast spectral optical coherence tomography,” Opt. Express 14(21), 9643–9656 (2006).
[Crossref] [PubMed]
G. Lan, M. Singh, K. V. Larin, and M. D. Twa, “Common-path phase-sensitive optical coherence tomography provides enhanced phase stability and detection sensitivity for dynamic elastography,” Biomed. Opt. Express 8(11), 5253–5266 (2017).
[Crossref] [PubMed]
J. Li, S. Wang, M. Singh, S. Aglyamov, S. Emelianov, M. Twa, and K. Larin, “Air-pulse OCE for assessment of age-related changes in mouse cornea in vivo,” Laser Phys. Lett. 11(6), 065601 (2014).
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[Crossref]
C. Wynne, “Two-mirror anastigmats,” JOSA B 59(5), 572–578 (1969).
[Crossref]
A. Gómez-Vieyra, A. Dubra, D. Malacara-Hernández, and D. R. Williams, “First-order design of off-axis reflective ophthalmic adaptive optics systems using afocal telescopes,” Opt. Express 17(21), 18906–18919 (2009).
[Crossref] [PubMed]
D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
Z. Han, S. R. Aglyamov, J. Li, M. Singh, S. Wang, S. Vantipalli, C. Wu, C. H. Liu, M. D. Twa, and K. V. Larin, “Quantitative assessment of corneal viscoelasticity using optical coherence elastography and a modified Rayleigh-Lamb equation,” J. Biomed. Opt. 20(2), 20501 (2015).
[Crossref] [PubMed]
M. Singh, J. Li, Z. Han, C. Wu, S. R. Aglyamov, M. D. Twa, and K. V. Larin, “Investigating elastic anisotropy of the porcine cornea as a function of intraocular pressure with optical coherence elastography,” J. Refract. Surg. 32(8), 562–567 (2016).
[Crossref] [PubMed]
M. Singh, J. Li, S. Vantipalli, S. Wang, Z. Han, A. Nair, S. R. Aglyamov, M. D. Twa, and K. V. Larin, “Noncontact elastic wave imaging optical coherence elastography for evaluating changes in corneal elasticity due to crosslinking,” IEEE J. Sel. Top. Quantum Electron. 22(3), 266–276 (2016).
[Crossref] [PubMed]
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[Crossref]

2018 (1)

S. Vantipalli, J. Li, M. Singh, S. R. Aglyamov, K. V. Larin, and M. D. Twa, “Effects of Thickness on Corneal Biomechanical Properties Using Optical Coherence Elastography,” Optom. Vis. Sci. 95(4), 299–308 (2018).
[Crossref] [PubMed]

2017 (1)

G. Lan, M. Singh, K. V. Larin, and M. D. Twa, “Common-path phase-sensitive optical coherence tomography provides enhanced phase stability and detection sensitivity for dynamic elastography,” Biomed. Opt. Express 8(11), 5253–5266 (2017).
[Crossref] [PubMed]

2016 (3)

M. Singh, J. Li, Z. Han, C. Wu, S. R. Aglyamov, M. D. Twa, and K. V. Larin, “Investigating elastic anisotropy of the porcine cornea as a function of intraocular pressure with optical coherence elastography,” J. Refract. Surg. 32(8), 562–567 (2016).
[Crossref] [PubMed]

M. Singh, J. Li, S. Vantipalli, S. Wang, Z. Han, A. Nair, S. R. Aglyamov, M. D. Twa, and K. V. Larin, “Noncontact elastic wave imaging optical coherence elastography for evaluating changes in corneal elasticity due to crosslinking,” IEEE J. Sel. Top. Quantum Electron. 22(3), 266–276 (2016).
[Crossref] [PubMed]

M. Singh, J. Li, Z. Han, S. Vantipalli, C.-H. Liu, C. Wu, R. Raghunathan, S. R. Aglyamov, M. D. Twa, and K. V. Larin, “Evaluating the effects of riboflavin/UV-A and Rose-Bengal/green light cross-linking of the rabbit cornea by noncontact optical coherence elastography,” Invest. Ophthalmol. Vis. Sci. 57(9), 112–120 (2016).
[Crossref] [PubMed]

2015 (2)

Z. Han, S. R. Aglyamov, J. Li, M. Singh, S. Wang, S. Vantipalli, C. Wu, C. H. Liu, M. D. Twa, and K. V. Larin, “Quantitative assessment of corneal viscoelasticity using optical coherence elastography and a modified Rayleigh-Lamb equation,” J. Biomed. Opt. 20(2), 20501 (2015).
[Crossref] [PubMed]

B. I. Akca, E. W. Chang, S. Kling, A. Ramier, G. Scarcelli, S. Marcos, and S. H. Yun, “Observation of sound-induced corneal vibrational modes by optical coherence tomography,” Biomed. Opt. Express 6(9), 3313–3319 (2015).
[Crossref] [PubMed]

2014 (3)

J. Li, S. Wang, M. Singh, S. Aglyamov, S. Emelianov, M. Twa, and K. Larin, “Air-pulse OCE for assessment of age-related changes in mouse cornea in vivo,” Laser Phys. Lett. 11(6), 065601 (2014).
[Crossref]

M. D. Twa, J. Li, S. Vantipalli, M. Singh, S. Aglyamov, S. Emelianov, and K. V. Larin, “Spatial characterization of corneal biomechanical properties with optical coherence elastography after UV cross-linking,” Biomed. Opt. Express 5(5), 1419–1427 (2014).
[Crossref] [PubMed]

J. Li, Z. Han, M. Singh, M. D. Twa, and K. V. Larin, “Differentiating untreated and cross-linked porcine corneas of the same measured stiffness with optical coherence elastography,” J. Biomed. Opt. 19(11), 110502 (2014).
[Crossref] [PubMed]

2013 (2)

S. Wang, K. V. Larin, J. Li, S. Vantipalli, R. K. Manapuram, S. Aglyamov, S. Emelianov, and M. D. Twa, “A focused air-pulse system for optical-coherence-tomography-based measurements of tissue elasticity,” Laser Phys. Lett. 10(7), 075605 (2013).
[Crossref] [PubMed]

S. Song, Z. Huang, T.-M. Nguyen, E. Y. Wong, B. Arnal, M. O’Donnell, and R. K. Wang, “Shear modulus imaging by direct visualization of propagating shear waves with phase-sensitive optical coherence tomography,” J. Biomed. Opt. 18(12), 121509 (2013).
[Crossref] [PubMed]

2012 (5)

J. Y. Hwang, S. Wachsmann-Hogiu, V. K. Ramanujan, J. Ljubimova, Z. Gross, H. B. Gray, L. K. Medina-Kauwe, and D. L. Farkas, “A multimode optical imaging system for preclinical applications in vivo: technology development, multiscale imaging, and chemotherapy assessment,” Mol. Imaging Biol. 14(4), 431–442 (2012).
[Crossref] [PubMed]

S. Wang, J. Li, R. K. Manapuram, F. M. Menodiado, D. R. Ingram, M. D. Twa, A. J. Lazar, D. C. Lev, R. E. Pollock, and K. V. Larin, “Noncontact measurement of elasticity for the detection of soft-tissue tumors using phase-sensitive optical coherence tomography combined with a focused air-puff system,” Opt. Lett. 37(24), 5184–5186 (2012).
[Crossref] [PubMed]

C. Dorronsoro, D. Pascual, P. Pérez-Merino, S. Kling, and S. Marcos, “Dynamic OCT measurement of corneal deformation by an air puff in normal and cross-linked corneas,” Biomed. Opt. Express 3(3), 473–487 (2012).
[Crossref] [PubMed]

C. Li, G. Guan, Z. Huang, M. Johnstone, and R. K. Wang, “Noncontact all-optical measurement of corneal elasticity,” Opt. Lett. 37(10), 1625–1627 (2012).
[Crossref] [PubMed]

C. Li, G. Guan, R. Reif, Z. Huang, and R. K. Wang, “Determining elastic properties of skin by measuring surface waves from an impulse mechanical stimulus using phase-sensitive optical coherence tomography,” J. R. Soc. Interface 9(70), 831–841 (2012).
[Crossref] [PubMed]

2011 (4)

C. Li, Z. Huang, and R. K. Wang, “Elastic properties of soft tissue-mimicking phantoms assessed by combined use of laser ultrasonics and low coherence interferometry,” Opt. Express 19(11), 10153–10163 (2011).
[Crossref] [PubMed]

A. Sarvazyan, T. J. Hall, M. W. Urban, M. Fatemi, S. R. Aglyamov, and B. S. Garra, “An overview of elastography-an emerging branch of medical imaging,” Curr. Med. Imaging Rev. 7(4), 255–282 (2011).
[Crossref] [PubMed]

M. R. Ford, W. J. Dupps, A. M. Rollins, A. S. Roy, and Z. Hu, “Method for optical coherence elastography of the cornea,” J. Biomed. Opt. 16(1), 016005 (2011).
[Crossref] [PubMed]

J. W. Ruberti, A. Sinha Roy, and C. J. Roberts, “Corneal biomechanics and biomaterials,” Annu. Rev. Biomed. Eng. 13(1), 269–295 (2011).
[Crossref] [PubMed]

2010 (1)

R. K. Wang and A. L. Nuttall, “Phase-sensitive optical coherence tomography imaging of the tissue motion within the organ of Corti at a subnanometer scale: a preliminary study,” J. Biomed. Opt. 15(5), 056005 (2010).
[Crossref] [PubMed]

2009 (2)

A. Gómez-Vieyra, A. Dubra, D. Malacara-Hernández, and D. R. Williams, “First-order design of off-axis reflective ophthalmic adaptive optics systems using afocal telescopes,” Opt. Express 17(21), 18906–18919 (2009).
[Crossref] [PubMed]

B. F. Kennedy, T. R. Hillman, R. A. McLaughlin, B. C. Quirk, and D. D. Sampson, “In vivo dynamic optical coherence elastography using a ring actuator,” Opt. Express 17(24), 21762–21772 (2009).
[Crossref] [PubMed]

2006 (3)

M. H. D. Torre-Ibarra, P. D. Ruiz, and J. M. Huntley, “Double-shot depth-resolved displacement field measurement using phase-contrast spectral optical coherence tomography,” Opt. Express 14(21), 9643–9656 (2006).
[Crossref] [PubMed]

S. J. Kirkpatrick, R. K. Wang, and D. D. Duncan, “OCT-based elastography for large and small deformations,” Opt. Express 14(24), 11585–11597 (2006).
[Crossref] [PubMed]

R. K. Wang, Z. Ma, and S. J. Kirkpatrick, “Tissue Doppler optical coherence elastography for real time strain rate and strain mapping of soft tissue,” Appl. Phys. Lett. 89(14), 144103 (2006).
[Crossref]

2005 (2)

K. Nariai and H. Iwamoto, “A variation of Schwarzschild telescope: golden section solution with two concentric spheres and its extension to finite distance solutions,” Opt. Rev. 12(3), 190–195 (2005).
[Crossref]

V. Y. Terebizh, “Two-mirror Schwarzschild aplanats: Basic relations,” Astron. Lett. 31(2), 129–139 (2005).
[Crossref]

2004 (2)

J. Rogowska, N. A. Patel, J. G. Fujimoto, and M. E. Brezinski, “Optical coherence tomographic elastography technique for measuring deformation and strain of atherosclerotic tissues,” Heart 90(5), 556–562 (2004).
[Crossref] [PubMed]

R. Chan, A. Chau, W. Karl, S. Nadkarni, A. Khalil, N. Iftimia, M. Shishkov, G. Tearney, M. Kaazempur-Mofrad, and B. Bouma, “OCT-based arterial elastography: robust estimation exploiting tissue biomechanics,” Opt. Express 12(19), 4558–4572 (2004).
[Crossref] [PubMed]

2003 (2)

J. F. Greenleaf, M. Fatemi, and M. Insana, “Selected methods for imaging elastic properties of biological tissues,” Annu. Rev. Biomed. Eng. 5(1), 57–78 (2003).
[Crossref] [PubMed]

K. Nightingale, S. McAleavey, and G. Trahey, “Shear-wave generation using acoustic radiation force: in vivo and ex vivo results,” Ultrasound Med. Biol. 29(12), 1715–1723 (2003).
[Crossref] [PubMed]

2001 (1)

A. Manduca, T. E. Oliphant, M. A. Dresner, J. L. Mahowald, S. A. Kruse, E. Amromin, J. P. Felmlee, J. F. Greenleaf, and R. L. Ehman, “Magnetic resonance elastography: non-invasive mapping of tissue elasticity,” Med. Image Anal. 5(4), 237–254 (2001).
[Crossref] [PubMed]

2000 (1)

Y. Zhao, Z. Chen, C. Saxer, S. Xiang, J. F. de Boer, and J. S. Nelson, “Phase-resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. 25(2), 114–116 (2000).
[Crossref] [PubMed]

1998 (2)

J. Schmitt, “OCT elastography: imaging microscopic deformation and strain of tissue,” Opt. Express 3(6), 199–211 (1998).
[Crossref] [PubMed]

C. L. de Korte, A. F. van der Steen, E. I. Céspedes, and G. Pasterkamp, “Intravascular ultrasound elastography in human arteries: initial experience in vitro,” Ultrasound Med. Biol. 24(3), 401–408 (1998).
[Crossref] [PubMed]

1997 (1)

A. Baranne and F. Launay, “Cassegrain: a famous unknown of instrumental astronomy,” J. Opt. 28(4), 158–172 (1997).
[Crossref]

1995 (1)

R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, “Magnetic resonance elastography by direct visualization of propagating acoustic strain waves,” Science 269(5232), 1854–1857 (1995).
[Crossref] [PubMed]

1994 (1)

R. Kingslake, “WHO? DISCOVERED CODDINGTON’S Equations?” Opt. Photonics News 5(8), 20–23 (1994).
[Crossref]

1991 (3)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: a quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13(2), 111–134 (1991).
[Crossref] [PubMed]

1980 (1)

M. Born, “E. Wolf Principles of optics,” Pergamon Press 6, 188–189 (1980).

1969 (1)

C. Wynne, “Two-mirror anastigmats,” JOSA B 59(5), 572–578 (1969).
[Crossref]

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Appl. Phys. Lett. (1)

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Curr. Med. Imaging Rev. (1)

Heart (1)

IEEE J. Sel. Top. Quantum Electron. (1)

Invest. Ophthalmol. Vis. Sci. (1)

J. Biomed. Opt. (5)

M. R. Ford, W. J. Dupps, A. M. Rollins, A. S. Roy, and Z. Hu, “Method for optical coherence elastography of the cornea,” J. Biomed. Opt. 16(1), 016005 (2011).
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Mol. Imaging Biol. (1)

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Ultrasound Med. Biol. (2)

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Fig. 1 Schematic of a classic Schwarzschild objective that consists of two mirrors. The object distance is demonstrated in a finite distance. The potential space (yellow shaded region) can be used to include additional channels (e.g. functional elements) to create a multi-channel imaging system.

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Fig. 2 Configuration geometry of the Schwarzschild scan objective which consists of concentric Mirror 1 and Mirror 2. Only one scanner was shown here for simplicity. (a) Ray-tracing for an arbitrary chief ray with the scan angle of θ. R₁ and R₂ are the radii of curvature, and T₁ and T₂ are the center thickness for Mirror 1 and Mirror 2, respectively. O is the mutual center point of the curvatures of the mirrors. S is the pivot point of the scanner. The distance between point S and Mirror 1 is d₁, the distance between Mirror 2 and Mirror 1 is d₂. The chief ray interacts with the two mirrors at points A and B with incident angles I₁ and I₂, respectively. The chief ray is incident on the focal plane (θ’) before the point C (exit pupil position). f is the focal length of the Schwarzschild scan object, and equals to the distance from point O to the focal plane. L is the total length. d_work and d_exp are the working distance and exit pupil distance, respectively. (b) Ray-tracing of the scan beams with the minimal and the maximal scan angles (θ_min and θ_max, respectively). The reserved central area of loading is in a central zone of ± H_min, while the peripheral area of scanning is in the annular zone of ± (H_min – H_max) at the focal plane. D₀ is the input beam size, D₁ is the diameter of Mirror 1, and D_{2_out} and D_{2_in} are the outer and inner diameters of Mirror 2, respectively. RH₁ to RH₅ are the specific marginal ray heights.

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Fig. 3 Axial dimensions of (a) total length L and (b) working distance d_work. To meet the requirements of L ≤ 300 mm and d_work ≥120 mm, R₁ should be in the range of 55 mm – 88.36 mm (yellow-shaded areas).

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Fig. 4 Marginal ray tracing for RH₁ to RH₅, to satisfy the dimensional constraint criteria in Eq. (7) for different values of R₁ and θ_min. (a) Satisfaction of first criterion, representing the minimum diameter for D_{2_out}. (b) Satisfaction of second criterion, representing the vignetting-free condition around the center hole in Mirror 2. (c) Satisfaction of the third criterion, representing the vignetting-free condition for Mirror 1 (ρ₁ = ρ_{2_out} = ρ_{2_in} = 90%). Yellow-shaded areas show the radial dimensional constraint requirements.

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Fig. 5 The equality 2RH₄/ρ₁ = 2RH₅ was used to determine (a) θ_min to θ_max, (b) H_min to H_max, (c) D₁ and D_{2_out}, and (d) OPD_max in related to the radius of the Mirror 1 (R₁). Yellow-shaded areas in (a) to (d) represent the desired value ranges. (e) Astigmatism for general scan angles (1° to 10°, with a step of 1°) without considering the physical constraints. (f) Astigmatism minimization using DMAA method (when α_i = 1) for the constraint-determined scan angle range of θ_min to θ_max.

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Fig. 6 RMS spot size reduction by employing an aspherical (k₁ = 0.436, solid lines) Mirror 1, compared to when Mirror 1 is spherical (k₁ = 0, dotted lines).

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Fig. 7 Schematic of the Schwarzschild scan objective, designed using the outcome of the design requirement analysis and Zemax software simulation (drawed in 3/4 section).

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Fig. 8 Distortion calibration in Zemax simulation. (a) Lateral distortion, demonstrated by the relation between scan angle and scan length. (b) Axial distortion, demonstrated by optical path difference (OPD) across the scan field.

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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.

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Equations (13)

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f = \frac{f_{1} f_{2}}{f + f_{2} - d_{2}} = \frac{M R_{1}}{2 (M - 1)},

d_{\exp} = [\frac{M d_{1} + M R_{1}}{2 d_{1} (M - 1) + R_{1} (M - 2)} + 1] \times R_{1} - T_{1},

d_{w o r k} = (\frac{M}{2 M - 2} + 1) \times R_{1} - T_{1},

d_{w o r k} = f + R_{1} - T_{1},

L = T_{1} + R_{2} + f = \frac{(2 M - 1)}{2 (M - 1)} \times M R_{1} + T_{1} .

{\begin{matrix} R H_{1} = (d_{1} - d_{2}) \tan θ_{\max} + \frac{D_{0}}{2}, \\ R H_{2} = [d_{1} (2 M - 1) + R_{1} (M - 1)] \times \tan θ_{\min} - \frac{(2 M - 1)}{2} D_{0}, \\ R H_{3} = [d_{1} (2 M - 1) + R_{1} (M - 1)] \times \tan θ_{\max} + \frac{(2 M - 1)}{2} D_{0}, \\ R H_{4} = d_{1} \tan θ_{\max} + \frac{D_{0}}{2}, \\ R H_{5} = {1 - \frac{[2 d_{1} (M - 1) + R_{1} (M - 2)] [R_{1} (M - 1) + T_{1}]}{M R_{1} [d_{1} (2 M - 1) + R_{1} (M - 1)]}} \\ \times {[d_{1} (2 M - 1) + R_{1} (M - 1)] \tan θ_{\min} - \frac{(2 M - 1)}{2} D_{0}}, \end{matrix}

{\begin{array}{l} \frac{2 R H_{3}}{ρ_{2_o u t}} \leq D_{2_o u t}, \\ 2 R H_{1} \leq D_{2_i n} \leq 2 R H_{2} \times ρ_{2_i n}, \\ \frac{2 R H_{4}}{ρ_{1}} \leq D_{1} \leq 2 R H_{5}, \end{array}

\begin{matrix} O P L (R_{1}, θ) = | \vec{S A} | + | \vec{A B} | + | \vec{B C} | \\ = \frac{\sin (I_{1} - θ) (\sin θ + \sin I_{2})}{\sin θ \sin I_{2}} \times R_{1} + \frac{1 + 2 (M - 1) \cos (2 I_{1} - I_{2} - θ)}{2 (M - 1) \cos (2 I_{1} - 2 I_{2} - θ)} \times M R_{1}, \end{matrix}

{\begin{matrix} I_{1} = arc \sin (\frac{d_{1} + R_{1}}{R_{1}} \times \sin θ), \\ I_{2} = arc \sin (\frac{d_{1} + R_{1}}{R_{2}} \times \sin θ) . \end{matrix}

O P D_{\max} (R_{1}, θ) = O P L (R_{1}, θ_{\max}) - O P L (R_{1}, θ_{\min}) .

A S T (R_{1}, 0) = \frac{M R_{1} (2 M - 2 + \cos I_{1}) \cos I_{2}}{4 (M - 1) + 2 \cos I_{1} - 2 M \cos I_{2}} - \frac{M R_{1} [(2 M - 1) + \sec I_{1}]}{2 \cos I_{2} [2 (M - 1) + \sec I_{1} - 2 M}

D M A A = \frac{\sum_{i = 0}^{k} a_{i} | A S T (R_{1}, θ_{i}) |}{\sum_{i = 0}^{k} a_{i}},

z (r) = \frac{r^{2} / R_{1}}{1 + \sqrt{1 - (1 + k_{1}) r^{2} / R_{1}^{2}}},