Abstract

We propose and demonstrate novel methods that enable simultaneous measurements of the phase index, the group index, and the geometrical thickness of an optically transparent object by combining optical low-coherence interferometer and confocal optics. The lowcoherence interferometer gives information relating the group index with the thickness, while the confocal optics allows access to the phase index related with the thickness of the sample. To relate these, two novel methods were devised. In the first method, the dispersion-induced broadening of the low-coherence envelop signal was utilized, and in the second method the frequency derivative of the phase index was directly obtained by taking the confocal measurements at several wavelengths. The measurements were made with eight different samples; B270, CaF2, two of BK7, two of fused silica, cover glass, and cigarette cover film. The average measurement errors of the first and the second methods were 0.123 % and 0.061 % in the geometrical thickness, 0.133 % and 0.066 % in the phase index, and 0.106 % and 0.057 % in the group index, respectively.

©2008 Optical Society of America

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References

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  1. D. F. Murphy and D. A. Flavin, “Dispersion-insensitive measurement of thickness and group index by low-coherence interferometry,” Appl. Opt. 39, 4607–4615 (2000).
    [Crossref]
  2. 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]
  3. 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] [PubMed]
  4. M. Ohmi, K. Yoden, Y. Ohnishi, and M. Haruna, “Optical tomography along the geometrical thickness by combination of coherence-gate and confocal imagings,” OFS 14, 288–291 (2000).
  5. T. Fukano and I. Yamaguchi, “Separation of measurement of the refractive index and the geometrical thickness by use of a wavelength-scanning interferometer with a confocal microscope,” Appl. Opt. 38, 4065–4073 (1999).
    [Crossref]
  6. A. V. Zvyagin, K. K. M. B. Dilusha Silva, S. A. Alexandrov, T. R. Hillman, J. J. Armstrong, T. Tsuzuki, and D. D. Sampson, “Refractive index tomography of turbid media by bifocal optical coherence refractometry,” Opt. Express,  11, 3503–3517 (2003).
    [Crossref] [PubMed]
  7. S. A. Alexandrov, A. V. Zvyagin, K. K. M. B. Dilusha Silva, and D. D. Sampson, “Bifocal optical coherence refractometry of turbid media,” Opt. Lett. 28, 117–119 (2003).
    [Crossref] [PubMed]
  8. M. Ohmi, Y. Ohnishi, K. Yoden, and M. Haruna, “In Vitro Simultaneous Measurement of Refractive Index and Thickness of Biological Tissue by the Low Coherence Interferometry,” IEEE TBME 47, 1266–1270 (2000).
  9. X. Wang, C. Zhang, L. Zhang, L. Wue, and J. Tian, “Simultaneous refractive index and thickness measurements of bio tissue by optical coherence tomography,” JBO 7, 628–632 (2002).
  10. M. Haruna, M. Ohmi, T. Mitsuyama, H. Tajiri, H. Maruyama, and M. Hashimoto, “Simultaneous measurement of the phase index and group indices and the thickness of transparent plates by low-coherence interferometry,” Opt. Lett. 23, 966–968 (1998).
    [Crossref]
  11. 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]
  12. H. Maruyama, S. Inoue, T. Mitsuyama, M. Ohmi, and M. Haruna, “Low-coherence interferometer system for the simultaneous measurement of refractive index and thickness,” Appl. Opt. 41, 1315–1322 (2002).
    [Crossref] [PubMed]
  13. M. Ohmi, H. Nishi, T. Konishi, 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]
  14. B. R. Masters, Confocal Microscopy and Multiphoton Excitation Microscopy (SPIE Press, 2005)
  15. B. E. Bouma and G. J. Tearney, Handbook of Optical Coherence Tomography (Marcel Dekker, Inc., 2002), Chap. 2.
  16. http://www.cvilaser.com/Common/PDFs/Dispersion_Equations.pdf
  17. M. Born and E. Wolf, Principles of Optics, 7 ed., (Cambridge university press, 1999) Ch. 10.

2004 (1)

M. Ohmi, H. Nishi, T. Konishi, 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 (2)

2002 (1)

2000 (3)

D. F. Murphy and D. A. Flavin, “Dispersion-insensitive measurement of thickness and group index by low-coherence interferometry,” Appl. Opt. 39, 4607–4615 (2000).
[Crossref]

M. Ohmi, Y. Ohnishi, K. Yoden, and M. Haruna, “In Vitro Simultaneous Measurement of Refractive Index and Thickness of Biological Tissue by the Low Coherence Interferometry,” IEEE TBME 47, 1266–1270 (2000).

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]

1999 (1)

1998 (1)

1996 (1)

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]

Alexandrov, S. A.

Armstrong, J. J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7 ed., (Cambridge university press, 1999) Ch. 10.

Bouma, B. E.

B. E. Bouma and G. J. Tearney, Handbook of Optical Coherence Tomography (Marcel Dekker, Inc., 2002), Chap. 2.

Flavin, D. A.

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]

Haruna, M.

M. Ohmi, H. Nishi, T. Konishi, 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, S. Inoue, T. Mitsuyama, M. Ohmi, and M. Haruna, “Low-coherence interferometer system for the simultaneous measurement of refractive index and thickness,” Appl. Opt. 41, 1315–1322 (2002).
[Crossref] [PubMed]

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. Ohmi, Y. Ohnishi, K. Yoden, and M. Haruna, “In Vitro Simultaneous Measurement of Refractive Index and Thickness of Biological Tissue by the Low Coherence Interferometry,” IEEE TBME 47, 1266–1270 (2000).

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

M. Ohmi, K. Yoden, Y. Ohnishi, and M. Haruna, “Optical tomography along the geometrical thickness by combination of coherence-gate and confocal imagings,” OFS 14, 288–291 (2000).

Hashimoto, M.

Hillman, T. R.

Inoue, S.

Konishi, T.

M. Ohmi, H. Nishi, T. Konishi, 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]

Maruyama, H.

Masters, B. R.

B. R. Masters, Confocal Microscopy and Multiphoton Excitation Microscopy (SPIE Press, 2005)

Mitsuyama, T.

Murphy, D. F.

Nishi, H.

M. Ohmi, H. Nishi, T. Konishi, 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, T. Konishi, 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, S. Inoue, T. Mitsuyama, M. Ohmi, and M. Haruna, “Low-coherence interferometer system for the simultaneous measurement of refractive index and thickness,” Appl. Opt. 41, 1315–1322 (2002).
[Crossref] [PubMed]

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. Ohmi, Y. Ohnishi, K. Yoden, and M. Haruna, “In Vitro Simultaneous Measurement of Refractive Index and Thickness of Biological Tissue by the Low Coherence Interferometry,” IEEE TBME 47, 1266–1270 (2000).

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

M. Ohmi, K. Yoden, Y. Ohnishi, and M. Haruna, “Optical tomography along the geometrical thickness by combination of coherence-gate and confocal imagings,” OFS 14, 288–291 (2000).

Ohnishi, Y.

M. Ohmi, Y. Ohnishi, K. Yoden, and M. Haruna, “In Vitro Simultaneous Measurement of Refractive Index and Thickness of Biological Tissue by the Low Coherence Interferometry,” IEEE TBME 47, 1266–1270 (2000).

M. Ohmi, K. Yoden, Y. Ohnishi, and M. Haruna, “Optical tomography along the geometrical thickness by combination of coherence-gate and confocal imagings,” OFS 14, 288–291 (2000).

Sampson, D. D.

Silva, K. K. M. B. Dilusha

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]

Tajiri, H.

Tearney, G. J.

B. E. Bouma and G. J. Tearney, Handbook of Optical Coherence Tomography (Marcel Dekker, Inc., 2002), Chap. 2.

Tian, J.

X. Wang, C. Zhang, L. Zhang, L. Wue, and J. Tian, “Simultaneous refractive index and thickness measurements of bio tissue by optical coherence tomography,” JBO 7, 628–632 (2002).

Tsuzuki, T.

Wang, X.

X. Wang, C. Zhang, L. Zhang, L. Wue, and J. Tian, “Simultaneous refractive index and thickness measurements of bio tissue by optical coherence tomography,” JBO 7, 628–632 (2002).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7 ed., (Cambridge university press, 1999) Ch. 10.

Wue, L.

X. Wang, C. Zhang, L. Zhang, L. Wue, and J. Tian, “Simultaneous refractive index and thickness measurements of bio tissue by optical coherence tomography,” JBO 7, 628–632 (2002).

Yamada, Y.

M. Ohmi, H. Nishi, T. Konishi, 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.

Yoden, K.

M. Ohmi, Y. Ohnishi, K. Yoden, and M. Haruna, “In Vitro Simultaneous Measurement of Refractive Index and Thickness of Biological Tissue by the Low Coherence Interferometry,” IEEE TBME 47, 1266–1270 (2000).

M. Ohmi, K. Yoden, Y. Ohnishi, and M. Haruna, “Optical tomography along the geometrical thickness by combination of coherence-gate and confocal imagings,” OFS 14, 288–291 (2000).

Zhang, C.

X. Wang, C. Zhang, L. Zhang, L. Wue, and J. Tian, “Simultaneous refractive index and thickness measurements of bio tissue by optical coherence tomography,” JBO 7, 628–632 (2002).

Zhang, L.

X. Wang, C. Zhang, L. Zhang, L. Wue, and J. Tian, “Simultaneous refractive index and thickness measurements of bio tissue by optical coherence tomography,” JBO 7, 628–632 (2002).

Zvyagin, A. V.

Appl. Opt. (3)

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]

IEEE TBME (1)

M. Ohmi, Y. Ohnishi, K. Yoden, and M. Haruna, “In Vitro Simultaneous Measurement of Refractive Index and Thickness of Biological Tissue by the Low Coherence Interferometry,” IEEE TBME 47, 1266–1270 (2000).

Meas. Sci. Technol. (1)

M. Ohmi, H. Nishi, T. Konishi, 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. Express (1)

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]

Other (6)

B. R. Masters, Confocal Microscopy and Multiphoton Excitation Microscopy (SPIE Press, 2005)

B. E. Bouma and G. J. Tearney, Handbook of Optical Coherence Tomography (Marcel Dekker, Inc., 2002), Chap. 2.

http://www.cvilaser.com/Common/PDFs/Dispersion_Equations.pdf

M. Born and E. Wolf, Principles of Optics, 7 ed., (Cambridge university press, 1999) Ch. 10.

M. Ohmi, K. Yoden, Y. Ohnishi, and M. Haruna, “Optical tomography along the geometrical thickness by combination of coherence-gate and confocal imagings,” OFS 14, 288–291 (2000).

X. Wang, C. Zhang, L. Zhang, L. Wue, and J. Tian, “Simultaneous refractive index and thickness measurements of bio tissue by optical coherence tomography,” JBO 7, 628–632 (2002).

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Figures (4)

Fig. 1.
Fig. 1. Schematic of the experimental set-up based on a low-coherence interferometer and confocal optics. BS: beam splitter, OBJ1: objective lens for the sample arm, OBJ2: objective lens for the reference arm, RM: reference mirror, PH: pin hole, L: focusing lens, SLD: superluminescent diode.

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Fig. 2.
Fig. 2. The measured confocal signal (a) and the low-coherence interference signal (b) of a 142 µm thick cover glass. The confocal distance Δz was measured smaller than the low-coherence distance Δl.

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Fig. 3.
Fig. 3. Discrepancy between the envelopes of the interference signals resulted from the rear surface and the front surface of a sample. The dotted line is obtained with experiment and the solid one is the fitted curve. From the curve fitting, the dispersion parameter can be achieved.

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Fig. 4.
Fig. 4. The phase index of a fused silica sample obtained with virtual experiments done by the proposed two methods and its reference value (solid curve). We can see that the measurements using three light sources give better accuracy than the ones using only two sources.

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Tables (3)

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Table 1. The three measurands measured for 8 samples at three different center wavelengths.

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Table 2. The three variables of samples calculated with the three measurands in Table 1 and by using the proposed first method.

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Table 3. The three variables of samples calculated with the measurands in Table 1 and by using the proposed second method.

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Equations (19)

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Δ z = t × 1 − N A 2 n p 2 − N A 2 ≈ 1 n p
Δ l = t × n g
Δ ϕ ( ω ) = 2 { k s ( ω ) x s + k L ( ω ) t } − 2 k r ( ω ) x r
Δ ϕ ( ω ) ≈ ω 0 Δ τ p + ( ω − ω 0 ) Δ τ g + ( ω − ω 0 ) 2 2 τ c 2 .
Δ τ p = 2 Δ x c + 2 t c n p ( ω 0 ) ,
Δ τ g = 2 Δ x c + 2 t c { n p ( ω 0 ) + ω 0 d n p d ω ∣ ω 0 } .
τ c 2 = 2 t c { 2 d n p d ω ∣ ω 0 + ω 0 d 2 n p d ω 2 ∣ ω 0 } .
n g ( ω ) = n p ( ω ) + ω d n p ( ω ) d ω ,
Δ τ g ( ω 0 ) = 2 Δ x c + 2 t c n g ( ω 0 )
τ c 2 ( ω 0 ) = 2 t c d n g ( ω ) d ω ∣ ω 0 .
I ∝ 1 2 π Re [ ∫ − ∞ ∞ S ( ω ) exp [ i Δ ϕ ( ω ) ] d ω ] .
I ∝ σ τ 2 σ τ 4 + τ c 4 exp { − σ τ 2 2 ( σ τ 4 + τ c 4 ) Δ τ g 2 } exp [ i { ω 0 Δ τ p − τ c 2 2 ( σ τ 4 + τ c 4 ) Δ τ g 2 + θ 0 } ] .
n p ( ω ) = Δ l Δ z ω { d ( Δ l Δ z ) d ω − c 2 τ c 2 Δ z } + Δ l Δ z .
t 2 = Δ l Δ z 1 − ω Δ z d Δ z d ω
A t 6 + B t 4 + C t 2 + D = 0
A = ( 1 − N A 2 ) 2 Δ z 4 + ω 2 ( 1 − N A 2 ) 2 Δ z 6 ( d Δ z d ω ) 2 − 2 ω ( 1 − N A 2 ) 2 Δ z 5 d Δ z d ω
B = 2 ( 1 − N A 2 ) Δ z 2 N A 2 − 2 N A 2 ω ( 1 − N A 2 ) Δ z 3 d Δ z d ω
C = N A 4 − Δ l 2 Δ z 2 ( 1 − N A 2 )
D = − Δ l 2 N A 2

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