Abstract

We report an analytical model of signal formation in spectrometer-based two-beam spectral interferometry. Considering the pixel size, the optical resolution and the spectral resolution of the spectrometer, and dispersion, the model represents the signal recorded by a spectrometer based on a diffraction grating and linear detector array. The model is general, but degenerates to more familiar forms with simplifying assumptions. The model is validated by comparison with experimental measurements, where it is shown that the model can accurately predict both signal fall-off and axial resolution for Fourier-domain optical coherence tomography imaging. The model may be useful for determining design specifications and expected performance parameters for spectrometers for spectral interferometry.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  24. S. H. Yun, G. J. Tearney, B. E. Bouma, B. H. Park, and J. F. d. Boer, "High-speed spectral-domain optical coherence tomography at 1.3 μm wavelength," Opt. Express 11, 3598-3604 (2003).
    [CrossRef] [PubMed]

2007

R. K. Wang, "In vivo full range complex Fourier domain optical coherence tomography," Appl. Phys. Lett. 90, 054103 (2007).

2006

2005

2004

2003

2002

2000

1999

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

1998

F. Hausler and M. W. Lindmer, "Coherence radar and spectral radar--new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Un-arunyawee, and J. A. Izatt, "In vivo video rate optical coherence tomography," Opt. Express 3, 219-229 (1998).
[CrossRef] [PubMed]

1997

1995

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 11, 43-48 (1995).
[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, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Aoki, G.

Bajraszewski, T.

Belabas, N.

Berisha, F.

Boer, J. F. d.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Bouma, B. E.

Cense, B.

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] [PubMed]

Chen, T. C.

Chen, Z.

Chinn, S. R.

Choma, M. A.

Dorrer, C.

Drexler, W.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Duker, J. S.

El-Zaiat, S. Y.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 11, 43-48 (1995).
[CrossRef]

Endo, T.

Fercher, A. 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] [PubMed]

Fujimoto, J. G.

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] [PubMed]

Hausler, F.

F. Hausler and M. W. Lindmer, "Coherence radar and spectral radar--new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

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, 1178-1181 (1991).
[CrossRef] [PubMed]

Hitzenberger, C. K.

Hu, Z.

Z. Hu and A. M. Rollins, "Theory of two beam interference with arbitrary spectra," Opt. Express 14, 12751-12759 (2006).
[CrossRef] [PubMed]

Z. Hu, M. Zhao, J. A. Izatt, and A. M. Rollins, "Enhancement of FDOCT imaging range by sub-pixel spectral shifting," in Coherence on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications, Systems and Technologies 2005 (Optical Society of America, 2005), p. CFA7.

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] [PubMed]

Itoh, M.

Izatt, J. A.

Z. Hu, M. Zhao, J. A. Izatt, and A. M. Rollins, "Enhancement of FDOCT imaging range by sub-pixel spectral shifting," in Coherence on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications, Systems and Technologies 2005 (Optical Society of America, 2005), p. CFA7.

M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 11, 2183-2189 (2003).
[CrossRef] [PubMed]

A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Un-arunyawee, and J. A. Izatt, "In vivo video rate optical coherence tomography," Opt. Express 3, 219-229 (1998).
[CrossRef] [PubMed]

Joffre, M.

Kamp, G.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 11, 43-48 (1995).
[CrossRef]

Ko, T. H.

Kowalczyk, A.

Kulkarni, M. D.

Lasser, T.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Leitgeb, R.

Leitgeb, R. A.

Likforman, J.-P.

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] [PubMed]

Lindmer, M. W.

F. Hausler and M. W. Lindmer, "Coherence radar and spectral radar--new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

Makita, S.

Mujat, M.

Nassif, N. A.

Nelson, J. S.

Pan, Y.

Park, B. H.

Pierce, M. 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, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Rollins, A. M.

Z. Hu and A. M. Rollins, "Theory of two beam interference with arbitrary spectra," Opt. Express 14, 12751-12759 (2006).
[CrossRef] [PubMed]

Z. Hu, M. Zhao, J. A. Izatt, and A. M. Rollins, "Enhancement of FDOCT imaging range by sub-pixel spectral shifting," in Coherence on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications, Systems and Technologies 2005 (Optical Society of America, 2005), p. CFA7.

A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Un-arunyawee, and J. A. Izatt, "In vivo video rate optical coherence tomography," Opt. Express 3, 219-229 (1998).
[CrossRef] [PubMed]

Sarunic, M. V.

Schmetterer, L.

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] [PubMed]

Srinivasan, V. J.

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] [PubMed]

Swanson, E. A.

S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, "Optical coherence tomography using a frequency-tunable optical source," Opt. Lett. 22, 340-342 (1997).
[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, 1178-1181 (1991).
[CrossRef] [PubMed]

Tearney, G.

Tearney, G. J.

Un-arunyawee, R.

Wang, H.

Wang, R. K.

R. K. Wang, "In vivo full range complex Fourier domain optical coherence tomography," Appl. Phys. Lett. 90, 054103 (2007).

Wang, Z.

Wojtkowski, M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Yang, C.

Yasuno, Y.

Yatagai, T.

Yazdanfar, S.

Yuan, Z.

Yun, S. H.

Yun, S.-H.

Zhang, J.

Zhao, M.

Z. Hu, M. Zhao, J. A. Izatt, and A. M. Rollins, "Enhancement of FDOCT imaging range by sub-pixel spectral shifting," in Coherence on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications, Systems and Technologies 2005 (Optical Society of America, 2005), p. CFA7.

Appl. Opt.

Appl. Phys. Lett.

R. K. Wang, "In vivo full range complex Fourier domain optical coherence tomography," Appl. Phys. Lett. 90, 054103 (2007).

J. Biomed. Opt.

F. Hausler and M. W. Lindmer, "Coherence radar and spectral radar--new tools for dermatological diagnosis," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, "Measurement of intraocular distances by backscattering spectral interferometry," Opt. Commun. 11, 43-48 (1995).
[CrossRef]

Opt. Express

A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Un-arunyawee, and J. A. Izatt, "In vivo video rate optical coherence tomography," Opt. Express 3, 219-229 (1998).
[CrossRef] [PubMed]

R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, "Performance of fourier domain vs. time domain optical coherence tomography," Opt. Express 11, 889-894 (2003).
[CrossRef] [PubMed]

S. H. Yun, G. J. Tearney, B. E. Bouma, B. H. Park, and J. F. d. Boer, "High-speed spectral-domain optical coherence tomography at 1.3 μm wavelength," Opt. Express 11, 3598-3604 (2003).
[CrossRef] [PubMed]

M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 11, 2183-2189 (2003).
[CrossRef] [PubMed]

M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, "Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation," Opt. Express 12, 2404-2422 (2004).
[CrossRef] [PubMed]

B. Cense, N. A. Nassif, T. C. Chen, M. C. Pierce, S.-H. Yun, B. H. Park, B. E. Bouma, G. J. Tearney, and J. F. d. Boer, "Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography," Opt. Express 12, 2435-2447 (2004).
[CrossRef] [PubMed]

Z. Wang, Z. Yuan, H. Wang, and Y. Pan, "Increasing the imaging depth of spectral-domain OCT by using interpixel shift technique," Opt. Express 14, 7014-7023 (2006).
[CrossRef] [PubMed]

Z. Hu and A. M. Rollins, "Theory of two beam interference with arbitrary spectra," Opt. Express 14, 12751-12759 (2006).
[CrossRef] [PubMed]

B. H. Park, M. C. Pierce, B. Cense, S.-H. Yun, M. Mujat, G. J. Tearney, B. E. Bouma, and J. F. d. Boer, "Real-time fiber-based multi-functional spectral domain optical coherence tomography at 1.3 μm," Opt. Express 13, 3931-3944 (2005).
[CrossRef] [PubMed]

Opt. Lett.

Rep. Prog. Phys.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

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, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Other

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Z. Hu, M. Zhao, J. A. Izatt, and A. M. Rollins, "Enhancement of FDOCT imaging range by sub-pixel spectral shifting," in Coherence on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications, Systems and Technologies 2005 (Optical Society of America, 2005), p. CFA7.

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Figures (6)

Fig. 1
Fig. 1

Schematic of the detector array illustrates the illumination by the spectrum and by the point spread function of an individual wavelength. Single wavelength PSF i covers multiple pixels; each rectangular pixel measures: width   Δ x × height   Δ y . Illuminating wavelength i is centered on pixel i. Pixel j is the pixel under investigation.

Fig. 2
Fig. 2

Plots of the bracketed factor in Eq. (3) representing the spectrometer resolution for a 50   μm pixel width and spot diameters a = 1 , 10, 25, 36, and 75   μm . The pixel function dominates when a = 1   μm , while the Gaussian PSF dominates when a = 75   μm .

Fig. 3
Fig. 3

Fall-off curves of conventional nonlinear-k FDOCT assuming varying and constant spot diameters. The simulated fall-off resulting from the variable spot size and from the constant effective spot size are almost identical.

Fig. 4
Fig. 4

Comparison of the simulated fall-off for grating-based nonlinear k spectral distribution (a), hypothetical linear k spectral distribution (b) and sinc function (c).

Fig. 5
Fig. 5

Fall-off curves at three different spot sizes a = 0 , 5, and 36   μm . The simulation for PSF width = 0   μm (diamonds), using the degenerate model in Eq. (7a), closely matches the simulation for PSF width = 5   μm (squares), using the full model in Eq. (4). The simulation for PSF width = 36   μm (triangles), using the full model in Eq. (4), closely matches the experimental measurement (circles) acquired using an FDOCT system equipped with a spectrometer with a 36   μm wide PSF.

Fig. 6
Fig. 6

FDOCT axial resolution as a function of path-length mismatch: experimentally measured resolution before (triangles) and after (squares) numerical second-order dispersion compensation, axial resolution measured from simulated data (circles), and simulated data with numerically introduced second-order dispersion mismatch (diamonds).

Equations (9)

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I ( x j ) = 0 0 s h ( x , y , k ) [ 1 2 ρ ref ( k ) + 1 2 ρ sam ( k ) + ρ ref ( k ) ρ sam ( k ) cos ( Δ ϕ ) ] d s d k .
h ( x , y , x i ) = 4 ln 2 π a 2 e ( 4 ln 2 / a 2 ) ( ( x x i ) 2 + y 2 ) ,
0 s h ( x , y , x i ) d s = x j Δ x / 2 x j + Δ x / 2 Δ y / 2 Δ y / 2 4 ln 2 π a 2 × e ( 4 ln 2 / a 2 ) ( ( x x i ) 2 + y 2 ) d y d x = 1 2   Erf ( Δ y ln   2 a ) × [ Erf ( ( Δ x 2 x i + 2 x j ) ln 2 a ) + Erf ( ( Δ x + 2 x i 2 x j ) ln 2 a ) ] ,
I ( x j ) = 1 4   Erf ( Δ y ln 2 a ) 0 [ Erf ( ( Δ x 2 x ( k ) + 2 x j ) ln 2 a ) + Erf ( ( Δ x + 2 x ( k ) 2 x j ) ln 2 a ) ] [ ρ ref ( k ) + ρ sam ( k ) + 2 ρ ref ( k ) ρ sam ( k ) cos ( Δ ϕ ) ] d k ,
A ( z ) = ( Δ x R ) e ( a 2 R 2 z 2 / 4 ln 2 ) sin ( Δ x R z ) Δ x R z × e [ ( z Δ L ) 2 σ 2 ln 2 / ln 2 2 + σ 4 ξ 2 ] ln 2 2 + σ 4 ξ 2 4 .
Axial   resolution = 2 ξ 2 σ 4 + ln ( 2 ) 2 σ ,
I ( x j ) = S ref ( k j ) + S sam ( k j ) + S ref ( k j ) S sam ( k j ) cos ( 2 k j Δ L ) × sin ( Δ k j Δ L ) Δ k j Δ L ( Δ x = finite ) ,
I ( x j ) = S ref ( k j ) + S sam ( k j ) + S ref ( k j ) S sam ( k j ) cos ( 2 k j Δ L ) ( Δ x 0 ) ,
x ( k ) = f [ sin 1 ( 2 π μ k π μ k c ) sin 1 ( 2 π μ k 0 π μ k c ) ] ,

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