Abstract

In this work we present a model for image formation in optical coherence microscopy. In the spectral domain detection, each wavenumber has a specific coherent transfer function that samples a different part of the object’s spatial frequency spectrum. The reconstruction of the tomogram is usually accurate only in a short depth of field. Using numerical simulations based on the developed model, we identified two distinct mechanisms that influence the signal of out-of-focus sample information. Besides the lateral blurring induced through defocusing, an additional axial envelope contributing equally to the signal degradation was found.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2009

K. M. Tan, M. Mazilu, T. H. Chow, W. M. Lee, K. Taguchi, B. K. Ng, W. Sibbett, C. S. Herrington, C. T. A. Brown, and K. Dholakia, “In-fiber common-path optical coherence tomography using a conical-tip fiber,” Opt. Express 17, 2375–2384 (2009).
[CrossRef] [PubMed]

M. Villiger, J. Goulley, M. Friedrich, A. Grapin-Botton, P. Meda, T. Lasser, and R. A. Leitgeb, “In vivo imaging of murine endocrine islets of langerhans with extended-focus optical coherence microscopy,” Diabetologia 52, 1599–1607 (2009).
[CrossRef] [PubMed]

2008

2007

2006

2005

2004

2003

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]

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]

2002

2000

M. Gu, Advanced Optical Imaging Theory, Vol. 75 of Springer Series in Optical Sciences (Springer, 2000).

1994

1964

1954

Adler, D. C.

Aguirre, A. D.

Akkin, T.

Bachmann, A. H.

Baumann, B.

Boppart, S. A.

Bouma, B.

Brezinski, M. E.

B. Liu and M. E. Brezinski, “Theoretical and practical considerations on detection performance of time domain, Fourier domain, and swept source optical coherence tomography,” J. Biomed. Opt. 12, 044007 (2007).
[CrossRef] [PubMed]

Brown, C. T. A.

Carney, P. S.

Cense, B.

Chen, N. G.

Chen, Z. P.

Choma, M. A.

Chow, T. H.

Coupland, J. M.

J. M. Coupland and J. Lobera, “Holography, tomography and 3d microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 074012 (2008).
[CrossRef]

Creazzo, T. L.

de Boer, J.

de Boer, J. E.

Dholakia, K.

Diaz, F.

Ding, Z. H.

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]

Ellerbee, A. K.

Fercher, A. F.

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]

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]

Friedrich, M.

M. Villiger, J. Goulley, M. Friedrich, A. Grapin-Botton, P. Meda, T. Lasser, and R. A. Leitgeb, “In vivo imaging of murine endocrine islets of langerhans with extended-focus optical coherence microscopy,” Diabetologia 52, 1599–1607 (2009).
[CrossRef] [PubMed]

Fujimoto, J. G.

Gotzinger, E.

Goulley, J.

M. Villiger, J. Goulley, M. Friedrich, A. Grapin-Botton, P. Meda, T. Lasser, and R. A. Leitgeb, “In vivo imaging of murine endocrine islets of langerhans with extended-focus optical coherence microscopy,” Diabetologia 52, 1599–1607 (2009).
[CrossRef] [PubMed]

Grapin-Botton, A.

M. Villiger, J. Goulley, M. Friedrich, A. Grapin-Botton, P. Meda, T. Lasser, and R. A. Leitgeb, “In vivo imaging of murine endocrine islets of langerhans with extended-focus optical coherence microscopy,” Diabetologia 52, 1599–1607 (2009).
[CrossRef] [PubMed]

Gu, M.

Guo, S. G.

Hee, M. R.

Herrington, C. S.

Hitzenberger, C. K.

Huang, S. W.

Huber, R. A.

Huignard, J. P.

Itoh, M.

Izatt, J. A.

Joo, C.

Kawata, S.

Kawata, Y.

Lasser, T.

Lee, K. S.

Lee, W. M.

Leitgeb, R.

Leitgeb, R. A.

Liu, B.

B. Liu and M. E. Brezinski, “Theoretical and practical considerations on detection performance of time domain, Fourier domain, and swept source optical coherence tomography,” J. Biomed. Opt. 12, 044007 (2007).
[CrossRef] [PubMed]

Liu, L. B.

Lobera, J.

J. M. Coupland and J. Lobera, “Holography, tomography and 3d microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 074012 (2008).
[CrossRef]

Loiseaux, B.

Makita, S.

Marks, D. L.

Mazilu, M.

McCutchen, C. W.

McLeod, J. H.

Meda, P.

M. Villiger, J. Goulley, M. Friedrich, A. Grapin-Botton, P. Meda, T. Lasser, and R. A. Leitgeb, “In vivo imaging of murine endocrine islets of langerhans with extended-focus optical coherence microscopy,” Diabetologia 52, 1599–1607 (2009).
[CrossRef] [PubMed]

Michaely, R.

Nakamura, Y.

Nelson, J. S.

Ng, B. K.

Owen, G. M.

Park, B. H.

Pircher, M.

Ralston, T. S.

Rao, B.

Ren, H. W.

Rolland, L. P.

Roy, M.

Sando, Y.

Sekhar, S. C.

Sharma, M. D.

Sheppard, C. J. R.

Sibbett, W.

Steinmann, L.

Su, J. P.

Sugisaka, J. I.

Swanson, E. A.

Taguchi, K.

Tan, K. M.

Tearney, G.

Tomlins, P. H.

P. H. Tomlins and R. K. Wang, “Theory, developments and applications of optical coherence tomography,” J. Phys. D: Appl. Phys. 38, 2519–2535 (2005).
[CrossRef]

Villiger, M.

M. Villiger, J. Goulley, M. Friedrich, A. Grapin-Botton, P. Meda, T. Lasser, and R. A. Leitgeb, “In vivo imaging of murine endocrine islets of langerhans with extended-focus optical coherence microscopy,” Diabetologia 52, 1599–1607 (2009).
[CrossRef] [PubMed]

R. A. Leitgeb, M. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier domain optical coherence microscopy,” Opt. Lett. 31, 2450–2452 (2006).
[CrossRef] [PubMed]

Wang, L.

Wang, Q.

Wang, R. K.

P. H. Tomlins and R. K. Wang, “Theory, developments and applications of optical coherence tomography,” J. Phys. D: Appl. Phys. 38, 2519–2535 (2005).
[CrossRef]

Yang, C. H.

Yasuno, Y.

Yatagai, T.

Yu, L. F.

Yun, S.

Zhang, J.

Zhao, Y. H.

Appl. Opt.

Diabetologia

M. Villiger, J. Goulley, M. Friedrich, A. Grapin-Botton, P. Meda, T. Lasser, and R. A. Leitgeb, “In vivo imaging of murine endocrine islets of langerhans with extended-focus optical coherence microscopy,” Diabetologia 52, 1599–1607 (2009).
[CrossRef] [PubMed]

J. Biomed. Opt.

B. Liu and M. E. Brezinski, “Theoretical and practical considerations on detection performance of time domain, Fourier domain, and swept source optical coherence tomography,” J. Biomed. Opt. 12, 044007 (2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D: Appl. Phys.

P. H. Tomlins and R. K. Wang, “Theory, developments and applications of optical coherence tomography,” J. Phys. D: Appl. Phys. 38, 2519–2535 (2005).
[CrossRef]

Meas. Sci. Technol.

J. M. Coupland and J. Lobera, “Holography, tomography and 3d microscopy as linear filtering operations,” Meas. Sci. Technol. 19, 074012 (2008).
[CrossRef]

Nat. Phys.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3, 129–134 (2007).
[CrossRef]

Opt. Express

Y. Yasuno, J. I. Sugisaka, Y. Sando, Y. Nakamura, S. Makita, M. Itoh, and T. Yatagai, “Non-iterative numerical method for laterally superresolving Fourier domain optical coherence tomography,” Opt. Express 14, 1006–1020 (2006).
[CrossRef] [PubMed]

L. F. Yu, B. Rao, J. Zhang, J. P. Su, Q. Wang, S. G. Guo, and Z. P. Chen, “Improved lateral resolution in optical coherence tomography by digital focusing using two-dimensional numerical diffraction method,” Opt. Express 15, 7634–7641 (2007).
[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. W. Huang, A. D. Aguirre, R. A. Huber, D. C. Adler, and J. G. Fujimoto, “Swept source optical coherence microscopy using a Fourier domain mode-locked laser,” Opt. Express 15, 6210–6217 (2007).
[CrossRef] [PubMed]

K. M. Tan, M. Mazilu, T. H. Chow, W. M. Lee, K. Taguchi, B. K. Ng, W. Sibbett, C. S. Herrington, C. T. A. Brown, and K. Dholakia, “In-fiber common-path optical coherence tomography using a conical-tip fiber,” Opt. Express 17, 2375–2384 (2009).
[CrossRef] [PubMed]

A. H. Bachmann, R. A. Leitgeb, and T. Lasser, “Heterodyne Fourier domain optical coherence tomography for full range probing with high axial resolution,” Opt. Express 14, 1487–1496 (2006).
[CrossRef] [PubMed]

B. Baumann, M. Pircher, E. Gotzinger, and C. K. Hitzenberger, “Full range complex spectral domain optical coherence tomography without additional phase shifters,” Opt. Express 15, 13375–13387 (2007).
[CrossRef] [PubMed]

S. Yun, G. Tearney, J. de Boer, and B. Bouma, “Removing the depth-degeneracy in optical frequency domain imaging with frequency shifting,” Opt. Express 12, 4822–4828 (2004).
[CrossRef] [PubMed]

Opt. Lett.

R. A. Leitgeb, R. Michaely, T. Lasser, and S. C. Sekhar, “Complex ambiguity-free Fourier domain optical coherence tomography through transverse scanning,” Opt. Lett. 32, 3453–3455 (2007).
[CrossRef] [PubMed]

R. A. Leitgeb, M. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier domain optical coherence microscopy,” Opt. Lett. 31, 2450–2452 (2006).
[CrossRef] [PubMed]

T. S. Ralston, D. L. Marks, S. A. Boppart, and P. S. Carney, “Inverse scattering for high-resolution interferometric microscopy,” Opt. Lett. 31, 3585–3587 (2006).
[CrossRef] [PubMed]

M. A. Choma, A. K. Ellerbee, C. H. Yang, T. L. Creazzo, and J. A. Izatt, “Spectral-domain phase microscopy,” Opt. Lett. 30, 1162–1164 (2005).
[CrossRef] [PubMed]

C. Joo, T. Akkin, B. Cense, B. H. Park, and J. E. de Boer, “Spectral-domain optical coherence phase microscopy for quantitative phase-contrast imaging,” Opt. Lett. 30, 2131–2133 (2005).
[CrossRef] [PubMed]

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, “Optical coherence microscopy in scattering media,” Opt. Lett. 19, 590–592 (1994).
[CrossRef] [PubMed]

Z. H. Ding, H. W. Ren, Y. H. Zhao, J. S. Nelson, and Z. P. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. 27, 243–245 (2002).
[CrossRef]

K. S. Lee and L. P. Rolland, “Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range,” Opt. Lett. 33, 1696–1698 (2008).
[CrossRef] [PubMed]

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]

Other

M. Gu, Advanced Optical Imaging Theory, Vol. 75 of Springer Series in Optical Sciences (Springer, 2000).

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Figures (8)

Fig. 1
Fig. 1

Layout of a spectrometer-based spectral domain OCT setup. Illumination and detection occurs through single mode fibers, and well defined illumination and detection modes. The free-space beam splitter decouples the detection from the illumination optics.

Fig. 2
Fig. 2

(a) Schematic of the situation at the sample objective. The field incident on the principal plane P 1 as a function of the radial coordinate ρ is transmitted by the objective to the principal plane P 2 , which has the shape of a sphere cap. The illumination field U ill is scattered by the susceptibility χ to produce the field U s . Only its overlap with the detection mode m det enters detection. (b) Principle of the generalized aperture. The plane wave components of the field in the focal plane are located in the spatial frequency domain on a sphere cap of radius k p . The angular distribution of the amplitude on this sphere cap corresponds to the angular distribution in the sample objective’s principal plane. (c) The three-dimensional Fourier transform of the generalized aperture produces the field in the focal region.

Fig. 3
Fig. 3

(a) Schematic of the convolution of two spherical shells, creating a circular overlap region. (b) CTF for identical generalized apertures with Gaussian modes and NA = 0.5 and κ = 0.7 . (c) The three-dimensional inverse Fourier transformation of the CTF produces the PSF ( k p = 2 π / λ p and λ p = 780   nm ).

Fig. 4
Fig. 4

(a) CTF for Gaussian illumination with NA = 0.5 and κ = 0.7 for three different wavenumbers. (b) Plot of the same three CTFs shown in (a), at q = 0 , evidencing the scaling of the CTF with k p . The amplitude decreases along with an scaling of the support length, making the integral of the CTF along s at q = 0 constant. (c) Simulated tomogram of idealized scatterers, aligned along the optical axis spaced by 10 μ m in logarithmic scale ( k p = 2 π / λ p and λ p = 780   nm , Δ k FWHM = k c / 5 ).

Fig. 5
Fig. 5

(a) Ideal CTF with very short support along s and no curvature. (b) Broad CTF, with separable q and s envelopes. (c) Cap CTF, with short s support, but curvature of radius 2 k p . (d), (e), (f) Tomograms produced with CTFs in (a), (b), and (c), respectively.

Fig. 6
Fig. 6

(a) Signal amplitude of idealized scatterers, positioned on the optical axis as a function of the out-of-focus distance, for the three idealized configurations presented in Fig. 5 and the realistic case in Fig. 4 at NA = 0.5 . (b) Signal amplitude at a given out-of-focus distance of 75 μ m for the same configurations as in (a), but with varying NA. (c) hwhm of the scatterer signals for several configurations as a function of the out-of-focus distance.

Fig. 7
Fig. 7

(a) Detailed view of out-of-focus scatterers for NA = 0.5 evidencing the axial curvature of the signal in the tomogram. (b) Comparison of the axial signal amplitude of the simulated scatterers with the envelope imposed by the PSF of the central wavenumber. Inset: Detailed view of the peak splitting of the scattering signal for various spectral widths. (c) Scattering signal in the lateral direction for the in-focus and out-of-focus ( z = 50 μ m ) situations, for the accurate model simulation and the direct PSF prediction. The legend applies to both (b) and (c).

Fig. 8
Fig. 8

(a)–(c) CTFs at the central wavenumber for (a) Bessel-like illumination and detection, (b) Bessel-like illumination with Gaussian detection, and (c) Gaussian illumination and detection, producing identical in-focus lateral resolutions of 1.3 μ m . (d) Projection of the same CTFs along the s direction, evidencing the transmission coefficients for the lateral spatial frequencies for the in-focus plane.

Equations (24)

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P tot ( k p ) = | S ref ( k p ) | 2 + | S s ( k p ) | 2 + S s ( k p ) S ref ( k p ) + S s ( k p ) S ref ( k p ) .
U ̃ ( r , k ) = i k   exp ( i k f ) 2 π P 2 U ̃ ( P 2 , k ) exp ( i k h ) h cos ( n , h ) d σ .
U ̃ ( r , k ) = i k f 2 π P 2 U ̃ ( p , k ) exp ( i k p r ) d Ω .
U ( r , t ) = i f 4 π 2 k P 2 U ̃ ( p , k ) k exp ( i c k t ) exp ( i k p r ) k 2 d k d Ω .
U ( r , t ) = i FT 3 D 1 { U ̃ ̃ ( Q ) exp ( i c Q t ) } .
U ( r , k p ) = i FT 3 D 1 { U ̃ ̃ ( Q , k p ) } ,
U ̃ ̃ ( Q , k p ) = i FT 3 D { U ( r , k p ) } .
U ̃ ̃ ( Q , k p ) = 2 π f U ̃ P 2 ( f q Q ) δ ( Q k p ) Q ,
U s ( r s , r 0 , k p ) = k p 2 4 π V U ill ( r , k p ) χ ( r r 0 ) exp ( i k p | r s r | ) | r s r | d V .
S s ( r 0 , k p ) = P 2 U s ( r s , r 0 , k p ) m det ( r s , k p ) d σ ,
S s ( r 0 , k p ) = k p 2 4 π V U ill ( r , k p ) χ ( r r 0 ) P 2 m det ( r s , k p ) exp ( i k p | r s r | ) | r s r | d σ d V .
P 2 m det ( r s , k p ) exp ( i k p | r s r | ) | r s r | d σ i 2 π   exp ( i k p f ) k p m det ( r , k p ) ,
S s ( r 0 , k p ) = i k p 2 V U ill ( r , k p ) m det ( r , k p ) χ ( r r 0 ) d V .
P ̃ ̃ s ( q , s , k p ) = i k p A ̃ ( k p ) 2 [ m ̃ ̃ ill ( q , s , k p ) m ̃ ̃ det ( q , s , k p ) ] χ ̃ ̃ ( q , s ) = i k p A ̃ ( k p ) CTF ( q , s , k p ) χ ̃ ̃ ( q , s ) .
CTF ( q , s ) = 4 π 2 f 2 β i β e m ̃ ill P 2 ( f q 1 k p ) m ̃ det P 2 ( f q 2 k p ) ξ Δ σ k p 2 d β = 4 π 2 f 2 Q β i β e m ̃ ill P 2 ( f q 1 k p ) m ̃ det P 2 ( f q 2 k p ) d β ,
CTF ( q , s ) = 4 π 2 f 2 Q s 1 i s 1 e m ̃ ill ( s 1 ) m ̃ det ( s s 1 ) d s 1 ξ 2 q 2 Q 2 ( s 1 s 2 ) 2 .
CTF [ n , m ] = 4 π 2 f 2 Q j = a b m ̃ ill [ j ] m ̃ det [ n j ] ( W + [ n , m , j ] W [ n , m , j ] ) ,
W ± [ n , m , j ] = sin 1 ( Q [ n , m ] ( s [ j ] ± Δ s 2 s [ n ] 2 ) q [ m ] ξ [ n , m ] ) ,
P ̃ s ( q , k p ) z = 0 = i k p A ̃ ( k p ) s CTF ( q , s , k p ) χ ̃ ̃ ( q , s ) d s .
T ( q , z ) = k p P ̃ s ( q , k p ) exp ( i 2 k p z ) d k p = i k p k p A ̃ ( k p ) s χ ̃ ̃ ( q , s ) CTF ( q , s , k p ) d s   exp ( i 2 k p z ) d k p .
T ( q , z ) = A ( z ) [ CTF ˜ ( q , z ) χ ̃ ( q , z ) ] .
T ( q , z ) = i k p k p A ̃ ( k p ) CTF ( q ) χ ̃ ̃ ( q , 2 k p ) exp ( i 2 k p z ) d k p .
χ ̃ ̃ ( q , s ) = n δ ( z n Δ z ) exp ( i Q r ) d Q = n exp ( i s n Δ z ) .
T ( r , z ) = PSF ( r , z ) [ A ( z ) n δ ( z n Δ z ) ] .

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