Abstract

Confocal scanning combined with low-coherence interferometry is used to provide remote refractive index and thickness measurements of transparent materials. The influence of lens aberrations in the confocal measurement is assessed through ray-trace modeling of the axial point-spread functions generated using optical configurations comprised of paired aspherics and paired achromats. Off-axis parabolic mirrors are suggested as an alternative to lenses and are shown to exhibit much more symmetric profiles provided the system numerical aperture is not too high. The modeled results compare favorably with experimental data generated using an optical instrument comprised of a broadband source and line-scan spectrometer. Refractive index and thickness measurements are made with each configuration with most mirror pairings offering better than twice the repeatability and accuracy of either lens pairing.

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References

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  1. B. L. Danielson and C. Y. Boisrobert, “Absolute optical ranging using low coherence interferometry,” Appl. Opt. 30, 2975–2979 (1991).
    [Crossref]
  2. 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, 1178–1181 (1991).
    [Crossref]
  3. W. V. Sorin and D. F. Gray, “Simultaneous thickness and group index measurement using optical low-coherence reflectometry,” IEEE Photon. Technol. Lett. 4, 105–107 (1992).
    [Crossref]
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    [Crossref]
  5. T. Fukano and I. Yamaguchi, “Simultaneous measurement of thickness and refractive indices of multiple layers by a low-coherence confocal interference microscope,” Opt. Lett. 21, 1942–1944 (1996).
    [Crossref]
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    [Crossref]
  7. H. Maruyama, T. Mitsuyama, M. Ohmi, and M. Haruna, “Simultaneous measurement of refractive index and thickness by low coherence interferometry considering chromatic dispersion of index,” Opt. Rev. 7, 468–472 (2000).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  16. D. Iwaniuk, P. Rastogi, and E. Hack, “Correcting spherical aberrations induced by an unknown medium through determination of its refractive index and thickness,” Opt. Express 19, 19407–19414 (2011).
    [Crossref]
  17. K. Newman, “An introduction to off-axis parabolic mirrors,” 2013, https://wp.optics.arizona.edu/optomech/wpcontent/uploads/sites/53/2016/10/521_Tutorial_Newman_Kevin.PDF .
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    [Crossref]
  19. Y. X. Yuan, “A review of trust region algorithms for optimization,” ICIAM 99, 271–282 (2011).
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    [Crossref]
  21. P. Liu, R. M. Groves, and R. Benedictus, “Signal processing in optical coherence tomography for aerospace material characterization,” Opt. Eng. 52, 032201 (2013).
    [Crossref]
  22. M. N. Polyanskiy, “Refractive index database,” 2019, https://refractiveindex.info .

2018 (1)

2013 (1)

P. Liu, R. M. Groves, and R. Benedictus, “Signal processing in optical coherence tomography for aerospace material characterization,” Opt. Eng. 52, 032201 (2013).
[Crossref]

2011 (2)

2009 (1)

J. J. Stamnes and D. Velauthapillai, “Focal shifts through a plane interface,” Opt. Commun. 282, 2286–2291 (2009).
[Crossref]

2008 (1)

2004 (1)

M. Ohmi, H. Nishi, Y. Konisha, Y. Yamada, and M. Haruna, “High-speed simultaneous measurement of refractive index and thickness of transparent plates by low-coherence interferometry and confocal optics,” Meas. Sci. Technol. 15, 1531–1535 (2004).
[Crossref]

2003 (1)

T. G. van Leeuwen, D. J. Faber, and M. C. Aalders, “Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography,” IEEE J. Sel. Top. Quantum 9, 227–233 (2003).
[Crossref]

2002 (1)

2000 (1)

H. Maruyama, T. Mitsuyama, M. Ohmi, and M. Haruna, “Simultaneous measurement of refractive index and thickness by low coherence interferometry considering chromatic dispersion of index,” Opt. Rev. 7, 468–472 (2000).
[Crossref]

1998 (1)

1997 (1)

C. J. R. Sheppard and P. Török, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).
[Crossref]

1996 (2)

1995 (1)

1993 (1)

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

1992 (1)

W. V. Sorin and D. F. Gray, “Simultaneous thickness and group index measurement using optical low-coherence reflectometry,” IEEE Photon. Technol. Lett. 4, 105–107 (1992).
[Crossref]

1991 (2)

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, 1178–1181 (1991).
[Crossref]

B. L. Danielson and C. Y. Boisrobert, “Absolute optical ranging using low coherence interferometry,” Appl. Opt. 30, 2975–2979 (1991).
[Crossref]

1988 (1)

Aalders, M. C.

T. G. van Leeuwen, D. J. Faber, and M. C. Aalders, “Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography,” IEEE J. Sel. Top. Quantum 9, 227–233 (2003).
[Crossref]

Benedictus, R.

P. Liu, R. M. Groves, and R. Benedictus, “Signal processing in optical coherence tomography for aerospace material characterization,” Opt. Eng. 52, 032201 (2013).
[Crossref]

Boisrobert, C. Y.

Bouma, B. E.

Brezinsky, M. E.

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, 1178–1181 (1991).
[Crossref]

Cremer, C.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

Danielson, B. L.

Diasporo, A.

Faber, D. J.

T. G. van Leeuwen, D. J. Faber, and M. C. Aalders, “Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography,” IEEE J. Sel. Top. Quantum 9, 227–233 (2003).
[Crossref]

Federici, F.

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, 1178–1181 (1991).
[Crossref]

Ford, H. D.

Francis, D.

Fujimoto, J. G.

G. J. Tearney, M. E. Brezinsky, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto, “Determination of the refractive index of highly scattering human tissue by optical coherence tomography,” Opt. Lett. 20, 2258–2260 (1995).
[Crossref]

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, 1178–1181 (1991).
[Crossref]

Fukano, T.

Gray, D. F.

W. V. Sorin and D. F. Gray, “Simultaneous thickness and group index measurement using optical low-coherence reflectometry,” IEEE Photon. Technol. Lett. 4, 105–107 (1992).
[Crossref]

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, 1178–1181 (1991).
[Crossref]

Groves, R. M.

P. Liu, R. M. Groves, and R. Benedictus, “Signal processing in optical coherence tomography for aerospace material characterization,” Opt. Eng. 52, 032201 (2013).
[Crossref]

Hack, E.

Haruna, M.

M. Ohmi, H. Nishi, Y. Konisha, Y. Yamada, and M. Haruna, “High-speed simultaneous measurement of refractive index and thickness of transparent plates by low-coherence interferometry and confocal optics,” Meas. Sci. Technol. 15, 1531–1535 (2004).
[Crossref]

H. Maruyama, T. Mitsuyama, M. Ohmi, and M. Haruna, “Simultaneous measurement of refractive index and thickness by low coherence interferometry considering chromatic dispersion of index,” Opt. Rev. 7, 468–472 (2000).
[Crossref]

M. Haruna, M. Ohmi, T. Misuyama, H. Tajiri, H. Maruyama, and M. Hashimoto, “Simultaneous measurement of the phase and group indices and the thickness of transparent plates by low-coherence interferometry,” Opt. Lett. 23, 966–968 (1998).
[Crossref]

Hashimoto, M.

Hee, M. R.

G. J. Tearney, M. E. Brezinsky, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto, “Determination of the refractive index of highly scattering human tissue by optical coherence tomography,” Opt. Lett. 20, 2258–2260 (1995).
[Crossref]

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, 1178–1181 (1991).
[Crossref]

Hell, S.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

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, 1178–1181 (1991).
[Crossref]

Iwaniuk, D.

Kim, M. J.

Kim, S.

Konisha, Y.

M. Ohmi, H. Nishi, Y. Konisha, Y. Yamada, and M. Haruna, “High-speed simultaneous measurement of refractive index and thickness of transparent plates by low-coherence interferometry and confocal optics,” Meas. Sci. Technol. 15, 1531–1535 (2004).
[Crossref]

Larkin, K.

Lee, B. H.

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, 1178–1181 (1991).
[Crossref]

Liu, P.

P. Liu, R. M. Groves, and R. Benedictus, “Signal processing in optical coherence tomography for aerospace material characterization,” Opt. Eng. 52, 032201 (2013).
[Crossref]

Maruyama, H.

H. Maruyama, T. Mitsuyama, M. Ohmi, and M. Haruna, “Simultaneous measurement of refractive index and thickness by low coherence interferometry considering chromatic dispersion of index,” Opt. Rev. 7, 468–472 (2000).
[Crossref]

M. Haruna, M. Ohmi, T. Misuyama, H. Tajiri, H. Maruyama, and M. Hashimoto, “Simultaneous measurement of the phase and group indices and the thickness of transparent plates by low-coherence interferometry,” Opt. Lett. 23, 966–968 (1998).
[Crossref]

Misuyama, T.

Mitsuyama, T.

H. Maruyama, T. Mitsuyama, M. Ohmi, and M. Haruna, “Simultaneous measurement of refractive index and thickness by low coherence interferometry considering chromatic dispersion of index,” Opt. Rev. 7, 468–472 (2000).
[Crossref]

Na, J.

Nishi, H.

M. Ohmi, H. Nishi, Y. Konisha, Y. Yamada, and M. Haruna, “High-speed simultaneous measurement of refractive index and thickness of transparent plates by low-coherence interferometry and confocal optics,” Meas. Sci. Technol. 15, 1531–1535 (2004).
[Crossref]

Ohmi, M.

M. Ohmi, H. Nishi, Y. Konisha, Y. Yamada, and M. Haruna, “High-speed simultaneous measurement of refractive index and thickness of transparent plates by low-coherence interferometry and confocal optics,” Meas. Sci. Technol. 15, 1531–1535 (2004).
[Crossref]

H. Maruyama, T. Mitsuyama, M. Ohmi, and M. Haruna, “Simultaneous measurement of refractive index and thickness by low coherence interferometry considering chromatic dispersion of index,” Opt. Rev. 7, 468–472 (2000).
[Crossref]

M. Haruna, M. Ohmi, T. Misuyama, H. Tajiri, H. Maruyama, and M. Hashimoto, “Simultaneous measurement of the phase and group indices and the thickness of transparent plates by low-coherence interferometry,” Opt. Lett. 23, 966–968 (1998).
[Crossref]

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, 1178–1181 (1991).
[Crossref]

Rastogi, P.

Reiner, G.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

Robello, M.

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, 1178–1181 (1991).
[Crossref]

Sheppard, C. J. R.

C. J. R. Sheppard and P. Török, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).
[Crossref]

C. J. R. Sheppard, “Aberrations in high aperture conventional and confocal imaging systems,” Appl. Opt. 27, 4782–4786 (1988).
[Crossref]

Sorin, W. V.

W. V. Sorin and D. F. Gray, “Simultaneous thickness and group index measurement using optical low-coherence reflectometry,” IEEE Photon. Technol. Lett. 4, 105–107 (1992).
[Crossref]

Southern, J. F.

Stamnes, J. J.

J. J. Stamnes and D. Velauthapillai, “Focal shifts through a plane interface,” Opt. Commun. 282, 2286–2291 (2009).
[Crossref]

Stelzer, E. H. K.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

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, 1178–1181 (1991).
[Crossref]

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, 1178–1181 (1991).
[Crossref]

Tajiri, H.

Tatam, R. P.

Tearney, G. J.

Török, P.

C. J. R. Sheppard and P. Török, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).
[Crossref]

van Leeuwen, T. G.

T. G. van Leeuwen, D. J. Faber, and M. C. Aalders, “Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography,” IEEE J. Sel. Top. Quantum 9, 227–233 (2003).
[Crossref]

Velauthapillai, D.

J. J. Stamnes and D. Velauthapillai, “Focal shifts through a plane interface,” Opt. Commun. 282, 2286–2291 (2009).
[Crossref]

Yamada, Y.

M. Ohmi, H. Nishi, Y. Konisha, Y. Yamada, and M. Haruna, “High-speed simultaneous measurement of refractive index and thickness of transparent plates by low-coherence interferometry and confocal optics,” Meas. Sci. Technol. 15, 1531–1535 (2004).
[Crossref]

Yamaguchi, I.

Yuan, Y. X.

Y. X. Yuan, “A review of trust region algorithms for optimization,” ICIAM 99, 271–282 (2011).

Appl. Opt. (3)

ICIAM (1)

Y. X. Yuan, “A review of trust region algorithms for optimization,” ICIAM 99, 271–282 (2011).

IEEE J. Sel. Top. Quantum (1)

T. G. van Leeuwen, D. J. Faber, and M. C. Aalders, “Measurement of the axial point spread function in scattering media using single-mode fiber-based optical coherence tomography,” IEEE J. Sel. Top. Quantum 9, 227–233 (2003).
[Crossref]

IEEE Photon. Technol. Lett. (1)

W. V. Sorin and D. F. Gray, “Simultaneous thickness and group index measurement using optical low-coherence reflectometry,” IEEE Photon. Technol. Lett. 4, 105–107 (1992).
[Crossref]

J. Microsc. (2)

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[Crossref]

C. J. R. Sheppard and P. Török, “Effects of specimen refractive index on confocal imaging,” J. Microsc. 185, 366–374 (1997).
[Crossref]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

M. Ohmi, H. Nishi, Y. Konisha, Y. Yamada, and M. Haruna, “High-speed simultaneous measurement of refractive index and thickness of transparent plates by low-coherence interferometry and confocal optics,” Meas. Sci. Technol. 15, 1531–1535 (2004).
[Crossref]

Opt. Commun. (1)

J. J. Stamnes and D. Velauthapillai, “Focal shifts through a plane interface,” Opt. Commun. 282, 2286–2291 (2009).
[Crossref]

Opt. Eng. (1)

P. Liu, R. M. Groves, and R. Benedictus, “Signal processing in optical coherence tomography for aerospace material characterization,” Opt. Eng. 52, 032201 (2013).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Opt. Rev. (1)

H. Maruyama, T. Mitsuyama, M. Ohmi, and M. Haruna, “Simultaneous measurement of refractive index and thickness by low coherence interferometry considering chromatic dispersion of index,” Opt. Rev. 7, 468–472 (2000).
[Crossref]

Science (1)

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, 1178–1181 (1991).
[Crossref]

Other (2)

K. Newman, “An introduction to off-axis parabolic mirrors,” 2013, https://wp.optics.arizona.edu/optomech/wpcontent/uploads/sites/53/2016/10/521_Tutorial_Newman_Kevin.PDF .

M. N. Polyanskiy, “Refractive index database,” 2019, https://refractiveindex.info .

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Figures (11)

Fig. 1.
Fig. 1. Refraction by a planar sample of thickness t; rays at angle θ1, where NA=sinθ1 are refocused from distance Δz inside the front surface onto the rear surface at t. θ2 is the refracted angle, and np and ng are the phase and group refractive indices, respectively.
Fig. 2.
Fig. 2. Focusing arrangements showing (a) aspheric lens pair with NA reduction, (b) achromat lens pair with NA increase, and (c) off-axis parabolic mirror pair with no NA change.
Fig. 3.
Fig. 3. Axial PSFs through the focus from the Zemax model (red curves), and with Gaussian broadening only (gray curves), for a beam focused by (a)–(c) paired aspheres or (d)–(f) paired achromats, onto a 1 mm thick planar sample, such that the focus coincides with the front (right-hand peaks) or rear (left-hand peaks) surface. Lens combinations correspond to system names given in Table 1. The NA at sample is (a) 0.185, (b) 0.107, (c) 0.080, (d) 0.172, (e) 0.110, and (f) 0.079.
Fig. 4.
Fig. 4. Axial PSFs through the focus from the Zemax model for a beam focused by paired paraboloidal mirrors onto a 1 mm thick planar dielectric sample. Curves correspond to the focus on (a) rear and (b) front sample surfaces, with an NA of 0.212 at the sample.
Fig. 5.
Fig. 5. Axial PSFs through the focus from the Zemax model (red curves), and with Gaussian broadening only (gray curves), for a beam focused by off-axis parabolic mirrors onto the front (right-hand peaks) or rear (left-hand peaks) surface of a 1 mm thick planar sample. Focusing system details are given in Table 1; (a) OAP5, NA at sample 0.082; (b) OAP3, NA at sample 0.179; and (c) OAP1, NA at sample 0.269.
Fig. 6.
Fig. 6. Difference between Δz values from Zemax model and those calculated using Eq. (1), as a function of NA, using peak positions defined as (a) centroid (with threshold at 33% of peak maximum) and (b) maximum value.
Fig. 7.
Fig. 7. Schematic showing the configuration of the experimental system. SLD is the super-luminescent diode, BC is the broadband coupler, and OAP is the off-axis parabolic mirror. The blue shading indicates which components are mounted on the translation stages. The inset shows the configuration used when confocal measurements are made with lenses.
Fig. 8.
Fig. 8. Image of the beam profile recorded with the collimating OAP mirror in the incorrect orientation shows significant coma in part (a). When correctly oriented and collimated, the beam profile through a shear plate interferometer shows straight line fringes oriented parallel to an alignment groove etched into the diffuse screen of the interferometer in part (b).
Fig. 9.
Fig. 9. Experimentally obtained confocal peaks (thick red curves) with the fit to the APSF given in Eq. (7) (thin black curves). The peaks shown were obtained using paired achromats [(a) rear peak and (b) front peak] and using paired aspheres [(c) rear peak and (d) front peak]. The NA at the sample is 0.150 for the achromats and 0.121 for aspheres.
Fig. 10.
Fig. 10. Experimentally obtained confocal peaks (thick red curves) with the fit to the axial PSF given in Eq. (7) (thin black curves) for different OAP mirror configurations. In each case the collimating optic is the same (f=25.4mm) with a different focusing optic in each case providing an NA at the sample of (a) and (b) 0.202; (c) and (d) 0.121; (e) and (f) 0.093; and (g) and (h) 0.061. The rear peaks are shown on the left, and the front peaks are shown on the right.
Fig. 11.
Fig. 11. Low coherence fringe envelopes obtained from the front sample surface using (a) the aspheric lens configuration and (b) the OAP mirror configuration with a 25.4 mm focuser. The variation in the confocal parameter Δz from which the dispersion data dΔz/dν is acquired (also obtained with OAP pairing with a 25.4 mm focuser).

Tables (3)

Tables Icon

Table 1. Summary of the Different Lens and Off-Axis Parabolic Mirror Configurations Used in the Assessmenta

Tables Icon

Table 2. Summary of the Properties of the Confocal Peaks Obtained with Different Optical Configurations (Shown in Figs. 9 and 10)a

Tables Icon

Table 3. Summary of Thickness and Refractive Index Measurements for a Range of Different Optical Configurations

Equations (8)

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Δz=t×nair2NA2np2NA2,
Δl=t×ng.
ng(ν)=np(ν)+νdnp(ν)dν,
t2=ΔlΔz1νΔzdΔzdν.
I(z)=I0(1exp(2r2/ω2(z)),
ω(z)=ω01+(z/zR)2,
h(d)=1(dz)2+1,
z=πnpωλ,