Abstract

Duct-profiling in test samples up to 25 mm in diameter has been demonstrated using a passive, low-coherence probe head with a depth resolution of 7.8 μm, incorporating an optical-fibre-linked conical mirror addressed by a custom-built array of single-mode fibres. Zemax modelling, and experimental assessment of instrument performance, show that degradation of focus, resulting from astigmatism introduced by the conical mirror, is mitigated by the introduction of a novel lens element. This enables a good beam focus to be achieved at distances of tens of millimetres from the cone axis, not achievable when the cone is used alone. Incorporation of the additional lens element is shown to provide a four-fold improvement in lateral imaging resolution, when compared with reflection from the conical mirror alone.

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References

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2013 (1)

H. D. Ford and R. P. Tatam, “Passive OCT probe head for 3D duct inspection,” Meas. Sci. Technol. 24(9), 094001 (2013).
[Crossref]

2011 (2)

H. D. Ford and R. P. Tatam, “Characterization of optical fiber imaging bundles for swept-source optical coherence tomography,” Appl. Opt. 50(5), 627–640 (2011).
[Crossref] [PubMed]

B. C. Quirk, R. A. McLaughlin, A. Curatolo, R. W. Kirk, P. B. Noble, and D. D. Sampson, “In situ imaging of lung alveoli with an optical coherence tomography needle probe,” J. Biomed. Opt. 16(3), 036009 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

R. Isago and K. Nakamura, “High-speed imaging with endoscopic optical coherence tomography using bending vibration of optical fiber,” Proc. SPIE 7503, 75034Y (2009).
[Crossref]

2007 (2)

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403 (2007).
[Crossref] [PubMed]

H. D. Ford and R. P. Tatam, “Fibre imaging bundles for full-field optical coherence tomography,” Meas. Sci. Technol. 18(9), 2949–2957 (2007).
[Crossref]

2006 (3)

I. Balboa, H. D. Ford, and R. P. Tatam, “Low-coherence optical fibre speckle interferometry,” Meas. Sci. Technol. 17(4), 605–616 (2006).
[Crossref]

H. D. Ford and R. P. Tatam, “Full-field optical coherence tomography using a fibre imaging bundle,” Proc. SPIE 6079, 60791H (2006).
[Crossref]

I. Durazo-Cardenas, P. Shore, X. Luo, T. Jacklin, S. A. Impey, and A. Cox, “3D characterisation of tool wear whilst diamond turning silicon,” Wear 262(3–4), 340–349 (2006).

2005 (1)

2004 (2)

2003 (1)

2002 (1)

M. de Rosa, J. Carberry, V. Bhagavatula, K. Wagner, and C. Saravanos, “High-power performance of single-mode fibre-optic connectors,” J. Lightwave Technol. 20(5), 879–885 (2002).
[Crossref]

2000 (1)

M. V. Sivak, K. Kobayashi, J. A. Izatt, A. M. Rollins, R. Ung-Runyawee, A. Chak, R. C. Wong, G. A. Isenberg, and J. Willis, “High-resolution endoscopic imaging of the GI tract using optical coherence tomography,” Gastrointest. Endosc. 51(4), 474–479 (2000).
[Crossref] [PubMed]

1996 (1)

Y.-J. Rao and D. A. Jackson, “Recent progress in fibre optic low-coherence interferometry,” Meas. Sci. Technol. 7(7), 981–999 (1996).
[Crossref]

1991 (2)

B. L. Danielson and C. Y. Boisrobert, “Absolute optical ranging using low coherence interferometry,” Appl. Opt. 30(21), 2975–2979 (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]

Alexandrov, S.

Armstrong, J.

Bajraszewski, T.

Balboa, I.

I. Balboa, H. D. Ford, and R. P. Tatam, “Low-coherence optical fibre speckle interferometry,” Meas. Sci. Technol. 17(4), 605–616 (2006).
[Crossref]

Bhagavatula, V.

M. de Rosa, J. Carberry, V. Bhagavatula, K. Wagner, and C. Saravanos, “High-power performance of single-mode fibre-optic connectors,” J. Lightwave Technol. 20(5), 879–885 (2002).
[Crossref]

Boisrobert, C. Y.

Boppart, S. A.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403 (2007).
[Crossref] [PubMed]

Bouma, B. E.

Brenner, M.

Carberry, J.

M. de Rosa, J. Carberry, V. Bhagavatula, K. Wagner, and C. Saravanos, “High-power performance of single-mode fibre-optic connectors,” J. Lightwave Technol. 20(5), 879–885 (2002).
[Crossref]

Chak, A.

M. V. Sivak, K. Kobayashi, J. A. Izatt, A. M. Rollins, R. Ung-Runyawee, A. Chak, R. C. Wong, G. A. Isenberg, and J. Willis, “High-resolution endoscopic imaging of the GI tract using optical coherence tomography,” Gastrointest. Endosc. 51(4), 474–479 (2000).
[Crossref] [PubMed]

Chang, W.

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]

Chen, Z.

Cox, A.

I. Durazo-Cardenas, P. Shore, X. Luo, T. Jacklin, S. A. Impey, and A. Cox, “3D characterisation of tool wear whilst diamond turning silicon,” Wear 262(3–4), 340–349 (2006).

Curatolo, A.

B. C. Quirk, R. A. McLaughlin, A. Curatolo, R. W. Kirk, P. B. Noble, and D. D. Sampson, “In situ imaging of lung alveoli with an optical coherence tomography needle probe,” J. Biomed. Opt. 16(3), 036009 (2011).
[Crossref] [PubMed]

Danielson, B. L.

de Rosa, M.

M. de Rosa, J. Carberry, V. Bhagavatula, K. Wagner, and C. Saravanos, “High-power performance of single-mode fibre-optic connectors,” J. Lightwave Technol. 20(5), 879–885 (2002).
[Crossref]

Drexler, W.

Durazo-Cardenas, I.

I. Durazo-Cardenas, P. Shore, X. Luo, T. Jacklin, S. A. Impey, and A. Cox, “3D characterisation of tool wear whilst diamond turning silicon,” Wear 262(3–4), 340–349 (2006).

Eastwood, P.

Fercher, A.

Flotte, T.

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]

Ford, H. D.

H. D. Ford and R. P. Tatam, “Passive OCT probe head for 3D duct inspection,” Meas. Sci. Technol. 24(9), 094001 (2013).
[Crossref]

H. D. Ford and R. P. Tatam, “Characterization of optical fiber imaging bundles for swept-source optical coherence tomography,” Appl. Opt. 50(5), 627–640 (2011).
[Crossref] [PubMed]

H. D. Ford and R. P. Tatam, “Fibre imaging bundles for full-field optical coherence tomography,” Meas. Sci. Technol. 18(9), 2949–2957 (2007).
[Crossref]

I. Balboa, H. D. Ford, and R. P. Tatam, “Low-coherence optical fibre speckle interferometry,” Meas. Sci. Technol. 17(4), 605–616 (2006).
[Crossref]

H. D. Ford and R. P. Tatam, “Full-field optical coherence tomography using a fibre imaging bundle,” Proc. SPIE 6079, 60791H (2006).
[Crossref]

Freilich, M. I.

Fujimoto, J. G.

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]

Gregory, K.

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]

Guo, S.

Hariri, L. P.

Hee, M. R.

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]

Hermann, B.

Hillman, D.

Huang, D.

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]

Impey, S. A.

I. Durazo-Cardenas, P. Shore, X. Luo, T. Jacklin, S. A. Impey, and A. Cox, “3D characterisation of tool wear whilst diamond turning silicon,” Wear 262(3–4), 340–349 (2006).

Isago, R.

R. Isago and K. Nakamura, “High-speed imaging with endoscopic optical coherence tomography using bending vibration of optical fiber,” Proc. SPIE 7503, 75034Y (2009).
[Crossref]

Isenberg, G. A.

M. V. Sivak, K. Kobayashi, J. A. Izatt, A. M. Rollins, R. Ung-Runyawee, A. Chak, R. C. Wong, G. A. Isenberg, and J. Willis, “High-resolution endoscopic imaging of the GI tract using optical coherence tomography,” Gastrointest. Endosc. 51(4), 474–479 (2000).
[Crossref] [PubMed]

Izatt, J. A.

M. V. Sivak, K. Kobayashi, J. A. Izatt, A. M. Rollins, R. Ung-Runyawee, A. Chak, R. C. Wong, G. A. Isenberg, and J. Willis, “High-resolution endoscopic imaging of the GI tract using optical coherence tomography,” Gastrointest. Endosc. 51(4), 474–479 (2000).
[Crossref] [PubMed]

Jacklin, T.

I. Durazo-Cardenas, P. Shore, X. Luo, T. Jacklin, S. A. Impey, and A. Cox, “3D characterisation of tool wear whilst diamond turning silicon,” Wear 262(3–4), 340–349 (2006).

Jackson, D. A.

Y.-J. Rao and D. A. Jackson, “Recent progress in fibre optic low-coherence interferometry,” Meas. Sci. Technol. 7(7), 981–999 (1996).
[Crossref]

Kirk, R. W.

B. C. Quirk, R. A. McLaughlin, A. Curatolo, R. W. Kirk, P. B. Noble, and D. D. Sampson, “In situ imaging of lung alveoli with an optical coherence tomography needle probe,” J. Biomed. Opt. 16(3), 036009 (2011).
[Crossref] [PubMed]

Kobayashi, K.

M. V. Sivak, K. Kobayashi, J. A. Izatt, A. M. Rollins, R. Ung-Runyawee, A. Chak, R. C. Wong, G. A. Isenberg, and J. Willis, “High-resolution endoscopic imaging of the GI tract using optical coherence tomography,” Gastrointest. Endosc. 51(4), 474–479 (2000).
[Crossref] [PubMed]

Le, T.

Leigh, M.

Leitgeb, R.

Lin, C. P.

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]

Luo, X.

I. Durazo-Cardenas, P. Shore, X. Luo, T. Jacklin, S. A. Impey, and A. Cox, “3D characterisation of tool wear whilst diamond turning silicon,” Wear 262(3–4), 340–349 (2006).

Marks, D. L.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403 (2007).
[Crossref] [PubMed]

McLaughlin, R. A.

B. C. Quirk, R. A. McLaughlin, A. Curatolo, R. W. Kirk, P. B. Noble, and D. D. Sampson, “In situ imaging of lung alveoli with an optical coherence tomography needle probe,” J. Biomed. Opt. 16(3), 036009 (2011).
[Crossref] [PubMed]

Mukai, D.

Mukai, D. S.

Nakamura, K.

R. Isago and K. Nakamura, “High-speed imaging with endoscopic optical coherence tomography using bending vibration of optical fiber,” Proc. SPIE 7503, 75034Y (2009).
[Crossref]

Nguyen, F. T.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403 (2007).
[Crossref] [PubMed]

Noble, P. B.

B. C. Quirk, R. A. McLaughlin, A. Curatolo, R. W. Kirk, P. B. Noble, and D. D. Sampson, “In situ imaging of lung alveoli with an optical coherence tomography needle probe,” J. Biomed. Opt. 16(3), 036009 (2011).
[Crossref] [PubMed]

Oh, W.-Y.

Oldenburg, A. L.

A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403 (2007).
[Crossref] [PubMed]

Puliafito, C. A.

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]

Quirk, B. C.

B. C. Quirk, R. A. McLaughlin, A. Curatolo, R. W. Kirk, P. B. Noble, and D. D. Sampson, “In situ imaging of lung alveoli with an optical coherence tomography needle probe,” J. Biomed. Opt. 16(3), 036009 (2011).
[Crossref] [PubMed]

Rao, Y.-J.

Y.-J. Rao and D. A. Jackson, “Recent progress in fibre optic low-coherence interferometry,” Meas. Sci. Technol. 7(7), 981–999 (1996).
[Crossref]

Rollins, A. M.

M. V. Sivak, K. Kobayashi, J. A. Izatt, A. M. Rollins, R. Ung-Runyawee, A. Chak, R. C. Wong, G. A. Isenberg, and J. Willis, “High-resolution endoscopic imaging of the GI tract using optical coherence tomography,” Gastrointest. Endosc. 51(4), 474–479 (2000).
[Crossref] [PubMed]

Rosenberg, M.

Sampson, D.

Sampson, D. D.

B. C. Quirk, R. A. McLaughlin, A. Curatolo, R. W. Kirk, P. B. Noble, and D. D. Sampson, “In situ imaging of lung alveoli with an optical coherence tomography needle probe,” J. Biomed. Opt. 16(3), 036009 (2011).
[Crossref] [PubMed]

Saravanos, C.

M. de Rosa, J. Carberry, V. Bhagavatula, K. Wagner, and C. Saravanos, “High-power performance of single-mode fibre-optic connectors,” J. Lightwave Technol. 20(5), 879–885 (2002).
[Crossref]

Schuman, J. S.

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]

Schwer, S.

Shishkov, M.

Shore, P.

I. Durazo-Cardenas, P. Shore, X. Luo, T. Jacklin, S. A. Impey, and A. Cox, “3D characterisation of tool wear whilst diamond turning silicon,” Wear 262(3–4), 340–349 (2006).

Sivak, M. V.

M. V. Sivak, K. Kobayashi, J. A. Izatt, A. M. Rollins, R. Ung-Runyawee, A. Chak, R. C. Wong, G. A. Isenberg, and J. Willis, “High-resolution endoscopic imaging of the GI tract using optical coherence tomography,” Gastrointest. Endosc. 51(4), 474–479 (2000).
[Crossref] [PubMed]

Stingl, A.

Stinson, W. G.

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]

Suter, M. J.

Swanson, E. A.

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]

Tatam, R. P.

H. D. Ford and R. P. Tatam, “Passive OCT probe head for 3D duct inspection,” Meas. Sci. Technol. 24(9), 094001 (2013).
[Crossref]

H. D. Ford and R. P. Tatam, “Characterization of optical fiber imaging bundles for swept-source optical coherence tomography,” Appl. Opt. 50(5), 627–640 (2011).
[Crossref] [PubMed]

H. D. Ford and R. P. Tatam, “Fibre imaging bundles for full-field optical coherence tomography,” Meas. Sci. Technol. 18(9), 2949–2957 (2007).
[Crossref]

I. Balboa, H. D. Ford, and R. P. Tatam, “Low-coherence optical fibre speckle interferometry,” Meas. Sci. Technol. 17(4), 605–616 (2006).
[Crossref]

H. D. Ford and R. P. Tatam, “Full-field optical coherence tomography using a fibre imaging bundle,” Proc. SPIE 6079, 60791H (2006).
[Crossref]

Tearney, G. J.

Tran, P. H.

Ung-Runyawee, R.

M. V. Sivak, K. Kobayashi, J. A. Izatt, A. M. Rollins, R. Ung-Runyawee, A. Chak, R. C. Wong, G. A. Isenberg, and J. Willis, “High-resolution endoscopic imaging of the GI tract using optical coherence tomography,” Gastrointest. Endosc. 51(4), 474–479 (2000).
[Crossref] [PubMed]

Unterhuber, A.

Vakoc, B. J.

Wagner, K.

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M. V. Sivak, K. Kobayashi, J. A. Izatt, A. M. Rollins, R. Ung-Runyawee, A. Chak, R. C. Wong, G. A. Isenberg, and J. Willis, “High-resolution endoscopic imaging of the GI tract using optical coherence tomography,” Gastrointest. Endosc. 51(4), 474–479 (2000).
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Biomed. Opt. Express (1)

Gastrointest. Endosc. (1)

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[Crossref] [PubMed]

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A. M. Zysk, F. T. Nguyen, A. L. Oldenburg, D. L. Marks, and S. A. Boppart, “Optical coherence tomography: a review of clinical development from bench to bedside,” J. Biomed. Opt. 12(5), 051403 (2007).
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Figures (14)

Fig. 1
Fig. 1

Conical mirror in typical geometry used for low-coherence duct inspection, showing two incident beams, each turned through 90° to address duct wall.

Fig. 2
Fig. 2

Side-on (left) and end-on (right) views of a 45° conical mirror of base radius R and height h, showing a unit vector n ^ normal to the cone surface and the Cartesian co-ordinate system used in beam reflection calculations.

Fig. 3
Fig. 3

Schematic 3D representation of, and diametric profile through, the custom cylindrical lens element. Not drawn to scale.

Fig. 4
Fig. 4

System geometry used in Zemax modelling to investigate beam focus for (a) conical mirror used alone, also showing the co-ordinate system used throughout, (b) custom lens and conical mirror in combination, with element separation distance defined. Beam centre in each case is 4mm from the cone axis, as shown.

Fig. 5
Fig. 5

(a) Best-resolution beam profiles, on a 1 mm square ‘virtual detector’, from the Zemax model (conical mirror alone), for a working distance of 15 mm, at three different NA values. (b) Beam waist radius at position of optimum compromise resolution, for conical mirror used alone. Symbols represent individual Zemax modelling values, with straight line fits to each data set, for four NA values from 0.015 to 0.1.

Fig. 6
Fig. 6

Plots from Zemax modelling, showing the evolution of axial and tangential beam waists as the custom lens/cone element combination is tracked parallel to the optical z-axis. Abscissae show distance from the original focus to the front surface of the custom lens. Plots assume a working distance of 12.5 mm, input NA of 0.06 and element separations of (a) 4.0 mm, (b) 2.8 mm, (c) 2.1 mm, (d) 1.5 mm, (e) 0.5 mm, (f) 0 mm. Legend in (a) applies to all plots.

Fig. 7
Fig. 7

Axial (squares) and tangential (diamonds) beam waist radii, at the position of optimum compromise resolution, from modelled profiles for conical mirror and custom lens element used together. Straight line fits to each data set are also shown, for NA values of 0.015 to 0.1.

Fig. 8
Fig. 8

Schematic of the optical arrangement, showing the broadband fibre coupler used for beam delivery and the bulk-optic low-coherence Michelson interferometer incorporating passive probe head. DC: directional coupler, VOA: Variable optical attenuator, BS: 80/20 beam-splitter, L: lens, M: mirror. Expanded 3D schematic below depicts optical probe-head components with example ray paths for a single fibre.

Fig. 9
Fig. 9

(a) Construction of 125 μm deep channel for laying up fibre array. (b) Channel loaded with fibres, inserted beneath the cover glass ‘roof’ and pushed back for equal overlap, showing notched plastc film used to apply pressure to close up gaps. (c) Photographic side view of curved array end on brass mandrel after assembly, potting and polishing. Fibres can be seen on upper surface of mandrel. (d) End-view microscope image showing portion of curved array face. (e) End-view composite of three light-microscope images, showing linear array face.

Fig. 10
Fig. 10

Detailed view of passive probe head, showing the 30 mm combined tube and cage system used to support the optical components.

Fig. 11
Fig. 11

(a) Centroid positions of fibres at output end of fibre array, from camera image of array output face. (b) Positions of selected array fibres with respect to cage-system axis, before (hollow markers) and after (solid markers) mandrel adjustment. The red dashed line shows a circle of radius 4mm, concentric with the mounting-system axis.

Fig. 12
Fig. 12

Experimental beam-waist radii at best focus for the low-coherence probe. Blue circles are values for the conical mirror used alone. Red markers are axial (squares) and tangential (diamonds) values for the conical mirror and custom lens element used together.

Fig. 13
Fig. 13

Schematic diagrams of tube-section samples used for testing the low-coherence profiling system. Radius of curvature of the test surface (shown in red) is (a) 8 mm, (b) 12.5 mm, (c) 18 mm and (d) 25 mm. (e) Photograph of the four 3D-printed samples.

Fig. 14
Fig. 14

Low-coherence surface depth profiles (blue points) from 3D-printed tube-section test samples having radii of curvature: (a) 8 mm, (b) 12.5 mm, (c) 18 mm and (d) 25 mm. Red dotted lines show the expected periodicity from theoretical calculation. (e)Plot of the measured depth information from (c) superimposed on a circle of radius 18 mm (blue solid line). The individual line segments from which the arc is constructed can be identified, and are highlighted by linear fits (red dotted lines) to the appropriate data subsets.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

n ^ = 1 2 (cosθ,sinθ,1).
r= i ^ 2( i ^ . n ^ ) n ^ .
r t =( 1,±( ( d r )1 )sinφ,0 ).
r a =(cosφ,0,sinφ)
1 f =(n1)( 1 R 1 1 R 2 ),

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