Abstract

We report a new method by which phase slope and group dispersion can be calculated with a simple optical low-coherence reflectometer to quantify physiological conditions. A discrete-time signal processing algorithm based on the first and second derivatives of the phase with respect to wave number was developed from discrete-time Fourier properties. The algorithm avoids the 2π ambiguity associated with most phase unwrapping. Experimental data collected by use of well-characterized optical materials validated the algorithm, which was minimally sensitive to phase noise. The group dispersion of bovine cornea was measured at various hydrations and was significantly dependent on hydration. The results suggest that group dispersion is an indicator of corneal alterations.

© 2003 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  32. P. J. Dougherty, K. L. Wellish, R. K. Maloney, “Excimer laser ablation rate and corneal hydration,” Am. J. Ophthalmol. 118, 169–176 (1994).
    [PubMed]

2003

2002

S. Jiao, L. H. Wang, “Two-dimensional depth-resolved Mueller matrix of biological tissue measured with double-beam polarization-sensitive optical coherence tomography,” Opt. Lett. 27, 101–103 (2002).
[CrossRef]

J. F. de Boer, T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359–371 (2002).
[CrossRef] [PubMed]

2001

2000

1999

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

C. K. Hitzenberger, A. Baumgartner, W. Drexler, A. F. Fercher, “Dispersion effects in partial coherence interferometry: implications for intraocular ranging,” J. Biomed. Opt. 4, 144–151 (1999).
[CrossRef] [PubMed]

1998

C. K. Hitzenberger, A. Baumgartner, A. F. Fercher, “Dispersion induced multiple signal peak splitting in partial coherence interferometry,” Opt. Commun. 154, 179–185 (1998).
[CrossRef]

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, A. Fercher, “Investigation of dispersion effects in ocular media by multiple wavelength partial coherence interferometry,” Exp. Eye Res. 66, 25–33 (1998).
[CrossRef] [PubMed]

1997

1996

A. F. Fercher, “Optical coherence tomography,” J. Biomed. Opt. 1, 157–173 (1996).
[CrossRef] [PubMed]

1995

1994

P. J. Dougherty, K. L. Wellish, R. K. Maloney, “Excimer laser ablation rate and corneal hydration,” Am. J. Ophthalmol. 118, 169–176 (1994).
[PubMed]

1992

M. R. Hee, D. Huang, E. A. Swanson, J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. B 9, 903–908 (1992).
[CrossRef]

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

1991

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

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

1966

B. Hedbys, S. Mishima, “The thickness-hydration relationship of the cornea,” Exp. Eye Res. 5, 221–228 (1966).
[CrossRef] [PubMed]

1965

I. Fatt, T. K. Goldstick, “Dynamics of water transport in swelling membranes,” J. Colloid. Sci. 20, 962–989 (1965).
[CrossRef] [PubMed]

Barton, J. K.

Baumgartner, A.

C. K. Hitzenberger, A. Baumgartner, W. Drexler, A. F. Fercher, “Dispersion effects in partial coherence interferometry: implications for intraocular ranging,” J. Biomed. Opt. 4, 144–151 (1999).
[CrossRef] [PubMed]

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, A. Fercher, “Investigation of dispersion effects in ocular media by multiple wavelength partial coherence interferometry,” Exp. Eye Res. 66, 25–33 (1998).
[CrossRef] [PubMed]

C. K. Hitzenberger, A. Baumgartner, A. F. Fercher, “Dispersion induced multiple signal peak splitting in partial coherence interferometry,” Opt. Commun. 154, 179–185 (1998).
[CrossRef]

Boisrobert, C. Y.

Boppart, S. A.

Bouma, B. E.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 189–194.

Brezinski, M. E.

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, Upper Saddle River, N.J., 1999).

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Chen, Z.

Danielson, B. L.

Dave, D. P.

de Boer, J. F.

J. F. de Boer, T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359–371 (2002).
[CrossRef] [PubMed]

J. F. de Boer, C. E. Saxer, J. S. Nelson, “Stable carrier generation and phase-resolved digital data processing in optical coherence tomography,” Appl. Opt. 40, 5787–5790 (2001).
[CrossRef]

Dougherty, P. J.

P. J. Dougherty, K. L. Wellish, R. K. Maloney, “Excimer laser ablation rate and corneal hydration,” Am. J. Ophthalmol. 118, 169–176 (1994).
[PubMed]

Draper, N. R.

N. R. Draper, H. Smith, Applied Regression Analysis (Wiley, New York, 1981).

Drexler, W.

U. Morgner, W. Drexler, X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25, 111–113 (2000).
[CrossRef]

C. K. Hitzenberger, A. Baumgartner, W. Drexler, A. F. Fercher, “Dispersion effects in partial coherence interferometry: implications for intraocular ranging,” J. Biomed. Opt. 4, 144–151 (1999).
[CrossRef] [PubMed]

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, A. Fercher, “Investigation of dispersion effects in ocular media by multiple wavelength partial coherence interferometry,” Exp. Eye Res. 66, 25–33 (1998).
[CrossRef] [PubMed]

Edney, P. A.

Fatt, I.

I. Fatt, T. K. Goldstick, “Dynamics of water transport in swelling membranes,” J. Colloid. Sci. 20, 962–989 (1965).
[CrossRef] [PubMed]

Fercher, A.

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, A. Fercher, “Investigation of dispersion effects in ocular media by multiple wavelength partial coherence interferometry,” Exp. Eye Res. 66, 25–33 (1998).
[CrossRef] [PubMed]

Fercher, A. F.

C. K. Hitzenberger, A. Baumgartner, W. Drexler, A. F. Fercher, “Dispersion effects in partial coherence interferometry: implications for intraocular ranging,” J. Biomed. Opt. 4, 144–151 (1999).
[CrossRef] [PubMed]

C. K. Hitzenberger, A. Baumgartner, A. F. Fercher, “Dispersion induced multiple signal peak splitting in partial coherence interferometry,” Opt. Commun. 154, 179–185 (1998).
[CrossRef]

A. F. Fercher, “Optical coherence tomography,” J. Biomed. Opt. 1, 157–173 (1996).
[CrossRef] [PubMed]

Findl, O.

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, A. Fercher, “Investigation of dispersion effects in ocular media by multiple wavelength partial coherence interferometry,” Exp. Eye Res. 66, 25–33 (1998).
[CrossRef] [PubMed]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

Goldstick, T. K.

I. Fatt, T. K. Goldstick, “Dynamics of water transport in swelling membranes,” J. Colloid. Sci. 20, 962–989 (1965).
[CrossRef] [PubMed]

Gray, D. F.

W. V. Sorin, 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Harris, T. J.

W. I. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 33.

Hedbys, B.

B. Hedbys, S. Mishima, “The thickness-hydration relationship of the cornea,” Exp. Eye Res. 5, 221–228 (1966).
[CrossRef] [PubMed]

Hee, M. E.

Hee, M. R.

Hitzenberger, C. K.

C. K. Hitzenberger, A. Baumgartner, W. Drexler, A. F. Fercher, “Dispersion effects in partial coherence interferometry: implications for intraocular ranging,” J. Biomed. Opt. 4, 144–151 (1999).
[CrossRef] [PubMed]

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, A. Fercher, “Investigation of dispersion effects in ocular media by multiple wavelength partial coherence interferometry,” Exp. Eye Res. 66, 25–33 (1998).
[CrossRef] [PubMed]

C. K. Hitzenberger, A. Baumgartner, A. F. Fercher, “Dispersion induced multiple signal peak splitting in partial coherence interferometry,” Opt. Commun. 154, 179–185 (1998).
[CrossRef]

Huang, D.

M. R. Hee, D. Huang, E. A. Swanson, J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. B 9, 903–908 (1992).
[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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Ippen, E. P.

Izatt, J. A.

Jiao, S.

Kärtner, X.

Kulkarni, M. D.

Li, X. D.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Maloney, R. K.

P. J. Dougherty, K. L. Wellish, R. K. Maloney, “Excimer laser ablation rate and corneal hydration,” Am. J. Ophthalmol. 118, 169–176 (1994).
[PubMed]

Marks, D. L.

Maurice, D. M.

D. M. Maurice, “The cornea and sclera,” in The Eye, H. Davson, ed. (Academic, Orlando, Fla., 1984), pp. 1–158.
[CrossRef]

Milner, T. E.

Mishima, S.

B. Hedbys, S. Mishima, “The thickness-hydration relationship of the cornea,” Exp. Eye Res. 5, 221–228 (1966).
[CrossRef] [PubMed]

Morgner, U.

Nelson, J. S.

Oldenburg, A. L.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, Upper Saddle River, N.J., 1999).

Pitris, C.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Reynolds, J. J.

Sattmann, H.

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, A. Fercher, “Investigation of dispersion effects in ocular media by multiple wavelength partial coherence interferometry,” Exp. Eye Res. 66, 25–33 (1998).
[CrossRef] [PubMed]

Saxer, C. E.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, Upper Saddle River, N.J., 1999).

Schmitt, J. M.

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

Schofield, M. A.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Smith, H.

N. R. Draper, H. Smith, Applied Regression Analysis (Wiley, New York, 1981).

Sorin, W. V.

W. V. Sorin, 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.

Steel, W. H.

W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), pp. 256–257.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

M. R. Hee, D. Huang, E. A. Swanson, J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. B 9, 903–908 (1992).
[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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tearney, G. J.

Thomas, C. W.

M. D. Kulkarni, C. W. Thomas, J. A. Izatt, “Image enhancement in optical coherence tomography using deconvolution,” Electron. Lett. 33, 1365–1367 (1997).
[CrossRef]

Thomas, M. E.

W. I. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 33.

Tropf, W. I.

W. I. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 33.

Walsh, J. T.

Wang, L. H.

Wang, X. J.

Welch, A. J.

Wellish, K. L.

P. J. Dougherty, K. L. Wellish, R. K. Maloney, “Excimer laser ablation rate and corneal hydration,” Am. J. Ophthalmol. 118, 169–176 (1994).
[PubMed]

Wexler, A.

A. Wexler, “Constant humidity solutions,” in Handbook of Chemistry and Physics, D. R. Lide, ed. (CRC, Boca Raton, Fla., 2000), Sec. 15–25.

Yazdanfar, S.

Zhu, Y.

Am. J. Ophthalmol.

P. J. Dougherty, K. L. Wellish, R. K. Maloney, “Excimer laser ablation rate and corneal hydration,” Am. J. Ophthalmol. 118, 169–176 (1994).
[PubMed]

Appl. Opt.

Electron. Lett.

M. D. Kulkarni, C. W. Thomas, J. A. Izatt, “Image enhancement in optical coherence tomography using deconvolution,” Electron. Lett. 33, 1365–1367 (1997).
[CrossRef]

Exp. Eye Res.

W. Drexler, C. K. Hitzenberger, A. Baumgartner, O. Findl, H. Sattmann, A. Fercher, “Investigation of dispersion effects in ocular media by multiple wavelength partial coherence interferometry,” Exp. Eye Res. 66, 25–33 (1998).
[CrossRef] [PubMed]

B. Hedbys, S. Mishima, “The thickness-hydration relationship of the cornea,” Exp. Eye Res. 5, 221–228 (1966).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron.

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quantum Electron. 5, 1205–1215 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

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

J. Biomed. Opt.

J. F. de Boer, T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359–371 (2002).
[CrossRef] [PubMed]

A. F. Fercher, “Optical coherence tomography,” J. Biomed. Opt. 1, 157–173 (1996).
[CrossRef] [PubMed]

C. K. Hitzenberger, A. Baumgartner, W. Drexler, A. F. Fercher, “Dispersion effects in partial coherence interferometry: implications for intraocular ranging,” J. Biomed. Opt. 4, 144–151 (1999).
[CrossRef] [PubMed]

J. Colloid. Sci.

I. Fatt, T. K. Goldstick, “Dynamics of water transport in swelling membranes,” J. Colloid. Sci. 20, 962–989 (1965).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

Opt. Commun.

C. K. Hitzenberger, A. Baumgartner, A. F. Fercher, “Dispersion induced multiple signal peak splitting in partial coherence interferometry,” Opt. Commun. 154, 179–185 (1998).
[CrossRef]

Opt. Lett.

Science

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Other

A. V. Oppenheim, R. W. Schafer, J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, Upper Saddle River, N.J., 1999).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 189–194.

A. Wexler, “Constant humidity solutions,” in Handbook of Chemistry and Physics, D. R. Lide, ed. (CRC, Boca Raton, Fla., 2000), Sec. 15–25.

W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983), pp. 256–257.

W. I. Tropf, M. E. Thomas, T. J. Harris, “Properties of crystals and glasses,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 33.

N. R. Draper, H. Smith, Applied Regression Analysis (Wiley, New York, 1981).

D. M. Maurice, “The cornea and sclera,” in The Eye, H. Davson, ed. (Academic, Orlando, Fla., 1984), pp. 1–158.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Experimental setup for the group-dispersion algorithm. Half of the light from the SLD goes to the reference arm; half goes to the sample arm. Reflected light from both arms interferes and is sensed at the detector. The interference signal formed by the light reflected from the coated mirror and the retroreflector is processed for calculating the group dispersion.

Fig. 2
Fig. 2

A scan of the microscope slide in the time domain. The lighter curve represents the interference signal, and the darker curve is the envelope of the signal obtained by quadrature-amplitude demodulation.

Fig. 3
Fig. 3

Phase slope dϕ/dk in the frequency domain. The shift in the phase slope at the central wavelength, λ = 820 nm, is the optical thickness of the microscope slide.

Fig. 4
Fig. 4

Interference signals from the coated mirror through three samples in the sample arm. Top, A scan without a sample. Coherence length l c is 18 μm, close to the theoretical coherence length of 16.5 μm. Center, A scan with the cuvette filled with water (10-mm inside thickness). Coherence length l c , is 22 μm. Bottom, A scan of a 2.08-mm-thick ZnSe disk. Coherence length l c is 41 μm.

Fig. 5
Fig. 5

Group dispersion of the cornea versus hydration and thickness. The error bars are 95% confidence intervals. Dashed curve, weighted least-squares regression. The intercept is 9.6(S.E. ± 4.0) × 10-5 nm-1, and the slope is -13(S.E. ± 4) × 10-11 m-2. The hydration is defined by the ratio of the water weight to the completely dry weight of the cornea.

Fig. 6
Fig. 6

Actual scanning speed distribution of 100 A scans. Each scanning speed was calculated from the peak of the spectral power density of each interference signal. Although the scanning speed was set at 16 mm/s, the actual speed varied as shown.

Equations (18)

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ϕk=tan-1ImFTInReFTIn,
eiw=|eiw|expi argeiw,
ˆeiwlog eiw=log|eiw|+i argeiw.
dˆeiwdw=dlog eiwdw=deiw/dweiw,
deiwdw=ddwn=- Inexp-iwn=-i n=- nInexp-iwn=-iFTnIn.
dϕdw=Imdˆeiwdw=Im-iFTnIneiw.
dϕdk=dϕdwdwdk=2V dϕdw,
dϕdk=2Vfsdϕdw=2VfsIm-iFTnIneiw,
dngdλ=-1z2πλ2d2ϕdk2,
d2ˆeiwdw2=ddwdlog eiwdw=ddwdeiw/dweiw=d2eiw/dw2eiw-deiw/dweiw2,
d2eiwdw2=d2dw2n=- Inexp-iwn=-n= n2Inexp-iwn=-FTn2In.
d2ˆeiwdw2=-FTn2Ineiw--iFTnIneiw2.
d2ϕdw2=Imd2ˆeiwdw2=Im-FTn2Ineiw--i FTnIneiw2.
d2ϕdk2=2Vfs2d2ϕdw2=2Vfs2Im-FTn2Ineiw--iFTnIneiw2.
dngdλ=-2πzλ22Vfs2Im-FTn2Ineiw--iFTnIneiw2.
dϕdkglass-air-dϕdkair-glass=Tng=1552 μm,
dngdλH2O=-1T2πλs2d2ϕdk2cuv-d2ϕdk2H2Oin cuv=-1.23S.D.±0.30×10-5nm-1,
dngdλZnSe=-1T2πλs2d2ϕdk2none-d2ϕdk2ZnSe disk=-38.58±1.23×10-5nm-1,

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