Abstract

We present a theoretical model to predict the sensitivity variation versus optical path difference (OPD) in Fourier domain spectral interferometry using configurations which produce Talbot bands. Such configurations require that the two interfering beams use different parts of the diffraction grating in the interrogating spectrometer. So far, the power distribution within the two beams in a Talbot bands experiment was considered uniform. In this report, we show that by manipulating the power distribution within the two interfering beams, the OPD value where maximum sensitivity is achieved can be conveniently tuned, as well as the sensitivity variation with OPD. Furthermore, creating a gap between the two beams leads to adjustment of the minimum detectable OPD value, while the width of the beams determine the maximum detectable OPD value. These features cannot be explained by theoretical models published so far involving spectrometer resolution elements only, while such features are correctly predicted by the model presented here.

© 2008 Optical Society of America

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  1. F. Talbot, "An experiment on the interference of light," Philos. Mag. 10, 364 (1837).
  2. A. L. King and R. Davis, "The Curious Bands of Talbot," American J. Phys. 39, 1195-1198 (1971).
    [CrossRef]
  3. G. B. Airy, "The Bakerian Lecture - on the theoretical explanation of an apparent new polarity in light," Phil. Trans. R. Soc. London 130, 225-244, (1840).
    [CrossRef]
  4. M. P. Givens "Talbot??s bands," Am. J. Phys. 61, 601-5 (1993).
    [CrossRef]
  5. R. S. Longhurst, Geometrical and Physical Optics (Longman, Inc., New York, 1973), Chap. 6.
  6. J. Jahns, A. W. Lohmann, and M. Bohling, "Talbots Bands and temporal processing of optical signals," J. Eur. Opt. Soc, Rapid Publ. 1, 06001 (2006).
    [CrossRef]
  7. Z. Benko, M. Hilbert, and Z. Bor, "New considerations on Talbot??s Bands," Am. J. Phys. 68, 513-520 (2000).
    [CrossRef]
  8. A. Gh. Podoleanu, "Unique interpretation of Talbot bands and Fourier domain white light interferometry," Opt. Express 15, 2007, 9867-9876
    [CrossRef] [PubMed]
  9. A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, " Talbot-like Bands for Laser Diode Below Threshold," J. Optics A: Pure Appl. Opt. 6, 413-424 (1997).
    [CrossRef]
  10. A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Theoretical Study of Talbot-like Bands Observed Using a Laser Diode Below Threshold," J. Optics A: Pure and Appl. Opt. 7, 517-536 (1998).
    [CrossRef]
  11. N. Nassif, B. Cense, B. Park, M. Pierce, S. Yun, B. Bouma, G. Tearney, T. Chen, and J. de Boer, "In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve," Opt. Express 12, 367-376 (2004).
    [CrossRef] [PubMed]
  12. Z. Hu, Y. Pan, and A. M. Rollins, "Analytical model of spectrometer-based two-beam spectral interferometry," Appl. Opt. 46, 8499-8505 (2007).
    [CrossRef] [PubMed]
  13. A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Channeled Spectrum Display using a CCD Array for Student Laboratory Demonstrations," Eur. J. Phys. 15, 266-271 (1994).
    [CrossRef]
  14. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, 1957), pp. 284
  15. L. M. Smith and C. C. Dobson, "Absolute Displacement Measurements using Modulation of the Spectrum of White Light in a Michelson Interferometer," Appl. Opt. 28, 3339-42 (1981).
    [CrossRef]
  16. J. Schwider and Liang Zhou, "Dispersive interferometric profilometer," Opt. Lett. 19, 995-997 (1994).
    [CrossRef] [PubMed]
  17. J. Zhang, J. S. Nelson, and Z. Chen, "Removal of a mirror image and enhancement of the signal-to-noise ratio in Fourier-domain optical coherence tomography using an electro-optic phase modulator," Opt. Lett. 30, 147-149 (2005).
    [CrossRef] [PubMed]
  18. A. Gh. Podoleanu and D. J. Woods "Power efficient FDOCT set-up for selection in the optical path difference sign using Talbot bands," Op. Lett. 32, 2300-2302 (2007).
    [CrossRef]
  19. L. M. Smith and C. C. Dobson, "Absolute Displacement Measurements using Modulation of the Spectrum of White Light in a Michelson Interferometer," Appl. Opt. 28, 3339-3342 (1981).
    [CrossRef]
  20. J. Schwider and Liang Zhou, "Dispersive interferometric profilometer," Opt. Lett. 19, 995-997 (1994).
    [CrossRef] [PubMed]
  21. K. -N. Joo and S. -W. Kim, "Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser," Opt. Express 14, 5954-5960 (2006).
    [CrossRef] [PubMed]
  22. S. Taplin, A. Gh. Podoleanu, D. J. Webb, and D. A. Jackson, "Displacement Sensor Using Channeled Spectrum Dispersed on a Linear CCD Array," Electron. Lett. 29, 896-897 (1993).
    [CrossRef]
  23. A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Channelled Spectrum Liquid Refractometer," Rev. Sci. Instr. 64, 3028-9 (1993).
    [CrossRef]
  24. M. W. Lindner, P. Andretzky, F. Kiesewetter, and G. Hausler, "Spectral radar: optical coherence tomography in the Fourier domain," Handbook of optical coherence tomography, B. E. Bouma and C. J. Tearney, eds., (Marcel Dekker Inc, New York-Basel, 2002), 335-358.
  25. S. H. Yun, G. J. Tearney, B. E. Bouma, B. H. Park, and J. F. de Boer, "High-speed spectral-domain optical coherence tomography at 1.3 um wavelength," Opt. Express 11, 3598-3604 (2003).
    [CrossRef] [PubMed]
  26. 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]
  27. J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, "Improved signal-tonoise ratio in spectral-domain compared with time-domain optical coherence tomography," Opt. Lett. 28, 2067-2069 (2003).
    [CrossRef] [PubMed]
  28. T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, "Profilometry with line-field Fourier-domain interferometry," Opt. Express 13, 695-701 (2005).
    [CrossRef] [PubMed]
  29. B. Park, M. C. Pierce, B. Cense, S. -H. Yun, M. Mujat, G. Tearney, B. Bouma, and J. de Boer, "Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 µm," Opt. Express 13, 3931-3944 (2005)
    [CrossRef] [PubMed]
  30. H. W. Lim and N. A. Soter, Clinical Photomedicine (Marcel Dekker, New York, 1993).
  31. M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
    [CrossRef] [PubMed]
  32. A. Bachmann, R. 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]
  33. A. B. Vakhtin, K. A. Peterson, and D. J. Kane, "Resolving the complex conjugate ambiguity in Fourier-domain OCT by harmonic lock-in detection of the spectral interferogram," Opt. Lett. 31, 1271-1273 (2006).
    [CrossRef] [PubMed]
  34. M. Sarunic, M. A. Choma, C. Yang, and J. A. Izatt, "Instantaneous complex conjugate resolved spectral domain and swept-source OCT using 3x3 fiber couplers," Opt. Express 13, 957-967 (2005).
    [CrossRef] [PubMed]

2007 (3)

2006 (4)

2005 (4)

2004 (1)

2003 (3)

2002 (1)

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

2000 (1)

Z. Benko, M. Hilbert, and Z. Bor, "New considerations on Talbot??s Bands," Am. J. Phys. 68, 513-520 (2000).
[CrossRef]

1998 (1)

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Theoretical Study of Talbot-like Bands Observed Using a Laser Diode Below Threshold," J. Optics A: Pure and Appl. Opt. 7, 517-536 (1998).
[CrossRef]

1997 (1)

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, " Talbot-like Bands for Laser Diode Below Threshold," J. Optics A: Pure Appl. Opt. 6, 413-424 (1997).
[CrossRef]

1994 (3)

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Channeled Spectrum Display using a CCD Array for Student Laboratory Demonstrations," Eur. J. Phys. 15, 266-271 (1994).
[CrossRef]

J. Schwider and Liang Zhou, "Dispersive interferometric profilometer," Opt. Lett. 19, 995-997 (1994).
[CrossRef] [PubMed]

J. Schwider and Liang Zhou, "Dispersive interferometric profilometer," Opt. Lett. 19, 995-997 (1994).
[CrossRef] [PubMed]

1993 (3)

S. Taplin, A. Gh. Podoleanu, D. J. Webb, and D. A. Jackson, "Displacement Sensor Using Channeled Spectrum Dispersed on a Linear CCD Array," Electron. Lett. 29, 896-897 (1993).
[CrossRef]

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Channelled Spectrum Liquid Refractometer," Rev. Sci. Instr. 64, 3028-9 (1993).
[CrossRef]

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Channelled Spectrum Liquid Refractometer," Rev. Sci. Instr. 64, 3028-9 (1993).
[CrossRef]

M. P. Givens "Talbot??s bands," Am. J. Phys. 61, 601-5 (1993).
[CrossRef]

1981 (2)

1971 (1)

A. L. King and R. Davis, "The Curious Bands of Talbot," American J. Phys. 39, 1195-1198 (1971).
[CrossRef]

1840 (1)

G. B. Airy, "The Bakerian Lecture - on the theoretical explanation of an apparent new polarity in light," Phil. Trans. R. Soc. London 130, 225-244, (1840).
[CrossRef]

1837 (1)

F. Talbot, "An experiment on the interference of light," Philos. Mag. 10, 364 (1837).

Airy, G. B.

G. B. Airy, "The Bakerian Lecture - on the theoretical explanation of an apparent new polarity in light," Phil. Trans. R. Soc. London 130, 225-244, (1840).
[CrossRef]

Bachmann, A.

Bajraszewski, T.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Benko, Z.

Z. Benko, M. Hilbert, and Z. Bor, "New considerations on Talbot??s Bands," Am. J. Phys. 68, 513-520 (2000).
[CrossRef]

Bohling, M.

J. Jahns, A. W. Lohmann, and M. Bohling, "Talbots Bands and temporal processing of optical signals," J. Eur. Opt. Soc, Rapid Publ. 1, 06001 (2006).
[CrossRef]

Bor, Z.

Z. Benko, M. Hilbert, and Z. Bor, "New considerations on Talbot??s Bands," Am. J. Phys. 68, 513-520 (2000).
[CrossRef]

Bouma, B.

Bouma, B. E.

Cense, B.

Chen, T.

Chen, Z.

Choma, M. A.

Davis, R.

A. L. King and R. Davis, "The Curious Bands of Talbot," American J. Phys. 39, 1195-1198 (1971).
[CrossRef]

de Boer, J.

de Boer, J. F.

Dobson, C. C.

Endo, T.

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]

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Gh, A.

A. Gh. Podoleanu, "Unique interpretation of Talbot bands and Fourier domain white light interferometry," Opt. Express 15, 2007, 9867-9876
[CrossRef] [PubMed]

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Theoretical Study of Talbot-like Bands Observed Using a Laser Diode Below Threshold," J. Optics A: Pure and Appl. Opt. 7, 517-536 (1998).
[CrossRef]

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, " Talbot-like Bands for Laser Diode Below Threshold," J. Optics A: Pure Appl. Opt. 6, 413-424 (1997).
[CrossRef]

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Channeled Spectrum Display using a CCD Array for Student Laboratory Demonstrations," Eur. J. Phys. 15, 266-271 (1994).
[CrossRef]

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Channelled Spectrum Liquid Refractometer," Rev. Sci. Instr. 64, 3028-9 (1993).
[CrossRef]

S. Taplin, A. Gh. Podoleanu, D. J. Webb, and D. A. Jackson, "Displacement Sensor Using Channeled Spectrum Dispersed on a Linear CCD Array," Electron. Lett. 29, 896-897 (1993).
[CrossRef]

Gh Podoleanu, A

A. Gh. Podoleanu and D. J. Woods "Power efficient FDOCT set-up for selection in the optical path difference sign using Talbot bands," Op. Lett. 32, 2300-2302 (2007).
[CrossRef]

Givens, M. P.

M. P. Givens "Talbot??s bands," Am. J. Phys. 61, 601-5 (1993).
[CrossRef]

Hilbert, M.

Z. Benko, M. Hilbert, and Z. Bor, "New considerations on Talbot??s Bands," Am. J. Phys. 68, 513-520 (2000).
[CrossRef]

Hitzenberger, C. K.

Hu, Z.

Itoh, M.

Izatt, J. A.

Jahns, J.

J. Jahns, A. W. Lohmann, and M. Bohling, "Talbots Bands and temporal processing of optical signals," J. Eur. Opt. Soc, Rapid Publ. 1, 06001 (2006).
[CrossRef]

Joo, K. -N.

Kane, D. J.

Kim, S. -W.

King, A. L.

A. L. King and R. Davis, "The Curious Bands of Talbot," American J. Phys. 39, 1195-1198 (1971).
[CrossRef]

Kowalczyk, A.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Lasser, T.

Leitgeb, R.

Lohmann, A. W.

J. Jahns, A. W. Lohmann, and M. Bohling, "Talbots Bands and temporal processing of optical signals," J. Eur. Opt. Soc, Rapid Publ. 1, 06001 (2006).
[CrossRef]

Makita, S.

Mujat, M.

Nassif, N.

Nelson, J. S.

Pan, Y.

Park, B.

Park, B. H.

Peterson, K. A.

Pierce, M.

Pierce, M. C.

Podoleanu, A.

S. Taplin, A. Gh. Podoleanu, D. J. Webb, and D. A. Jackson, "Displacement Sensor Using Channeled Spectrum Dispersed on a Linear CCD Array," Electron. Lett. 29, 896-897 (1993).
[CrossRef]

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Channelled Spectrum Liquid Refractometer," Rev. Sci. Instr. 64, 3028-9 (1993).
[CrossRef]

Rollins, A. M.

Sarunic, M.

Schwider, J.

Smith, L. M.

Talbot, F.

F. Talbot, "An experiment on the interference of light," Philos. Mag. 10, 364 (1837).

Taplin, S.

S. Taplin, A. Gh. Podoleanu, D. J. Webb, and D. A. Jackson, "Displacement Sensor Using Channeled Spectrum Dispersed on a Linear CCD Array," Electron. Lett. 29, 896-897 (1993).
[CrossRef]

Tearney, G.

Tearney, G. J.

Vakhtin, A. B.

Wojtkowski, M.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Woods, D. J.

A. Gh. Podoleanu and D. J. Woods "Power efficient FDOCT set-up for selection in the optical path difference sign using Talbot bands," Op. Lett. 32, 2300-2302 (2007).
[CrossRef]

Yang, C.

Yasuno, Y.

Yatagai, T.

Yun, S.

Yun, S. H.

Yun, S. -H.

Zhang, J.

Zhou, Liang

Am. J. Phys. (2)

M. P. Givens "Talbot??s bands," Am. J. Phys. 61, 601-5 (1993).
[CrossRef]

Z. Benko, M. Hilbert, and Z. Bor, "New considerations on Talbot??s Bands," Am. J. Phys. 68, 513-520 (2000).
[CrossRef]

American J. Phys. (1)

A. L. King and R. Davis, "The Curious Bands of Talbot," American J. Phys. 39, 1195-1198 (1971).
[CrossRef]

Appl. Opt. (3)

Electron. Lett. (1)

S. Taplin, A. Gh. Podoleanu, D. J. Webb, and D. A. Jackson, "Displacement Sensor Using Channeled Spectrum Dispersed on a Linear CCD Array," Electron. Lett. 29, 896-897 (1993).
[CrossRef]

Eur. J. Phys. (1)

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Channeled Spectrum Display using a CCD Array for Student Laboratory Demonstrations," Eur. J. Phys. 15, 266-271 (1994).
[CrossRef]

J. Biomed. Opt. (1)

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. F. Fercher, "In vivo human retinal imaging by Fourier domain optical coherence tomography," J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

J. Optics A: Pure and Appl. Opt. (1)

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Theoretical Study of Talbot-like Bands Observed Using a Laser Diode Below Threshold," J. Optics A: Pure and Appl. Opt. 7, 517-536 (1998).
[CrossRef]

J. Optics A: Pure Appl. Opt. (1)

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, " Talbot-like Bands for Laser Diode Below Threshold," J. Optics A: Pure Appl. Opt. 6, 413-424 (1997).
[CrossRef]

Op. Lett. (1)

A. Gh. Podoleanu and D. J. Woods "Power efficient FDOCT set-up for selection in the optical path difference sign using Talbot bands," Op. Lett. 32, 2300-2302 (2007).
[CrossRef]

Opt. Express (9)

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

N. Nassif, B. Cense, B. Park, M. Pierce, S. Yun, B. Bouma, G. Tearney, T. Chen, and J. de Boer, "In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve," Opt. Express 12, 367-376 (2004).
[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]

K. -N. Joo and S. -W. Kim, "Absolute distance measurement by dispersive interferometry using a femtosecond pulse laser," Opt. Express 14, 5954-5960 (2006).
[CrossRef] [PubMed]

A. Gh. Podoleanu, "Unique interpretation of Talbot bands and Fourier domain white light interferometry," Opt. Express 15, 2007, 9867-9876
[CrossRef] [PubMed]

T. Endo, Y. Yasuno, S. Makita, M. Itoh, and T. Yatagai, "Profilometry with line-field Fourier-domain interferometry," Opt. Express 13, 695-701 (2005).
[CrossRef] [PubMed]

M. Sarunic, M. A. Choma, C. Yang, and J. A. Izatt, "Instantaneous complex conjugate resolved spectral domain and swept-source OCT using 3x3 fiber couplers," Opt. Express 13, 957-967 (2005).
[CrossRef] [PubMed]

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

A. Bachmann, R. 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]

Opt. Lett. (5)

Phil. Trans. R. Soc. London (1)

G. B. Airy, "The Bakerian Lecture - on the theoretical explanation of an apparent new polarity in light," Phil. Trans. R. Soc. London 130, 225-244, (1840).
[CrossRef]

Philos. Mag. (1)

F. Talbot, "An experiment on the interference of light," Philos. Mag. 10, 364 (1837).

Rapid Publ. (1)

J. Jahns, A. W. Lohmann, and M. Bohling, "Talbots Bands and temporal processing of optical signals," J. Eur. Opt. Soc, Rapid Publ. 1, 06001 (2006).
[CrossRef]

Rev. Sci. Instr. (1)

A. Gh. Podoleanu, S. Taplin, D. J. Webb, and D. A. Jackson, "Channelled Spectrum Liquid Refractometer," Rev. Sci. Instr. 64, 3028-9 (1993).
[CrossRef]

Other (4)

M. W. Lindner, P. Andretzky, F. Kiesewetter, and G. Hausler, "Spectral radar: optical coherence tomography in the Fourier domain," Handbook of optical coherence tomography, B. E. Bouma and C. J. Tearney, eds., (Marcel Dekker Inc, New York-Basel, 2002), 335-358.

H. W. Lim and N. A. Soter, Clinical Photomedicine (Marcel Dekker, New York, 1993).

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, 1957), pp. 284

R. S. Longhurst, Geometrical and Physical Optics (Longman, Inc., New York, 1973), Chap. 6.

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Figures (10)

Fig. 1.
Fig. 1.

Generation of Talbot bands. CL: colimating lens; DG: diffraction grating, SL: spectrometer lens; A, A‵, A‵‵: positions where the glass slide can be inserted partially into the beam to produce Talbot Bands.

Fig. 2.
Fig. 2.

Spectrometer arrangement to produce Talbot Bands, consisting of a diffraction grating, a CCD array and a spectrometer lens, SL. Two collimated beams, one from the object arm and the other from the reference arm in an interferometer are directed onto different parts of the diffraction grating. These beams replace the two beams created by inserting the microscope slide into the incoming beam in Fig. 1. The power distribution into the two beams is described by profiles B1 and B2, The intrinsic delay across each beam is given in terms of delay D as defined in Eq. (2). The wavelets incident on and diffracted by each grating line in the two diffracted beams are shown for OPD=0 (no extrinsic delay). The spatial extention of the wavelets illustrates the coherence length of the incoming pulses. The diffracted beam is compounded from all individual diffracted wavelets as shown and therefore, the diffracted wavetrain exhibits a much longer length than that of the incoming beam. The extrinsic delay given by Eq. (6) is zero (pulses of spatial transversal profiles B1 and B2 arrive at the same time at the spectrometer, as shown in the left. In this sketch, diffracted wavetrain 2 incurs an intrinsic delay due to diffraction in relation to the diffracted wavetrain 1.

Fig. 3.
Fig. 3.

Left: Classical Michelson interferometer where the two beams are totally superposed. Both beams cover the same 5000 grating lines. Center: distribution of powers in the two beams is a top hat. Right: Visibility of channelled spectrum.

Fig. 4.
Fig. 4.

Left: Each beam is half obscured with an opaque screen, to mimic the original Talbot bands experiment. Center: distribution of powers in the two beams is top hat. Each beam covers 2500 grating lines. Right: Visibility of channelled spectrum which shows practically no interference for negative path difference (this equates to elimination of mirror terms) and exhibits a quasi-triangular shape with a peak at 2 mm (=λ0.2500). In accordance with the sketch in Fig. 1, channeled spectrum is obtained when wavetrain 1 is delayed extrinsically in relation to wavetrain 2, hence according to convention of sign established by Eq. (6), when OPD>0.

Fig. 5.
Fig. 5.

Top row: η=25%, bottom row η=75%; Left column: optical configuration; Middle column: profiles B1 and B2 which determine the distribution of power in the two beam; Right: channelled spectrum visibility.

Fig. 6.
Fig. 6.

Illustration of superposition of the two diffracted wavetrains, (1) and (2) for the two cases presented in Fig. 5. Each segment of the diffracted wavetrains contains 1250 individual wavelets. For η=25%, the diffracted wavetrains are 3 segments long=3750 wavelets and they are superposed for 2 segments in OPD=0, while for η=75% they are of one segment long and they are apart by 2 segments=2500 wavelets in OPD=0.

Fig. 7.
Fig. 7.

Left: A beam is split into two beams, beam 1 is restricted to m=2500 lines and beam 2 to M=1000 lines Center: distribution of powers in the two beams is top hat,. The beams are adjusted to have a gap of h=500 lines between them. The beams in this simulation are normalized so total power in each is equal. Right: Visibility of channelled spectrum. Note that the maximum visibility is less than 1.

Fig. 8.
Fig. 8.

Left: power efficient FD-OCT set-up to adjust the lateral gap between the two beams. Top right: Gaussian beam profiles separated by 2500 grating lines, and each beam covers approx. 2500 grating lines. Bottom right, visibility profile for Gaussian beams (solid line) and top-hat beams (dotted line) which is displayed for comparison.

Fig. 9:
Fig. 9:

Left: Gaussian beams apodized so that no overlap on the face of the grating occurs. Right: log representation of the corresponding visibility profile (solid line) in comparison with the log-representation of the non-apertured beams shown in Fig. 8 top right (dotted line). The asymmetry in the latter profile (exaggerated by the logarithmic representation) results from truncation of ‘tails’ of the Gaussian beam profiles at the edge of the diffraction grating.

Fig. 10.
Fig. 10.

Left: beam profiles for apodized Gaussian beams with a Gaussian shadow in beam 1. The case where σ=100,000 is shown only. Right: the resulting visibility for σ=10,000, σ=100,000 and σ=1,000,000.

Equations (18)

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V ( β , λ ¯ , t ) = 1 2 { z 1 ( D , λ ¯ ) L 1 ( λ ¯ ) X ( λ ¯ , t ) + Z 2 ( D , λ ¯ ) L 2 ( λ ¯ ) X ( λ ¯ , t ) }
D ( β ) = d ( sin α + sin β )
X ( λ ¯ , t ) = Y ( λ ¯ ) exp [ i 2 π c t λ ¯ ]
Z 1 = r = 1 m B 1 ( r ) exp ( i 2 π λ ¯ r D ) and Z 2 = r = m + h + 1 m + h + M B 2 ( r ) exp ( i 2 π λ ¯ r D )
L 1 ( λ ¯ ) X ( λ ¯ , t ) = T 1 exp [ i 2 π c ( t t 1 ) λ ¯ ]
L 2 ( λ ¯ ) X ( λ ¯ , t ) = T 2 exp [ i 2 π c ( t t 2 ) λ ¯ ]
Δ t = t 1 t 2 = OPD c
V V * ( D , λ ¯ , Δ t ) Y ( λ ¯ ) 2 { s = ( m 1 ) m 1 T 1 2 C 1 ( s ) . e i 2 π λ ¯ D s + s = ( M 1 ) M 1 T 2 2 C 2 ( s ) e i 2 π λ ¯ D s + 2 s = 1 ( h + m + M ) 1 h T 1 T 2 C 12 ( s ) cos [ 2 π λ ¯ ( c Δ t + s D ) ] }
C 12 ( s ) = r = h + m + 1 m + M + h B 1 ( r ) · B 2 ( r + s )
C 1 ( s ) = r = 1 m B 1 ( r ) · B 1 ( r + s ) and C 2 ( s ) = r = m + h + 1 m + M + h B 2 ( r ) · B 2 ( r + s )
g ( x ) = exp [ 2 ( π σ x ) 2 ]
ψ ( x ) = g ( x ) cos 2 π λ 0 ¯ x
I ( D ) s = 1 m m 1 T 1 2 C 1 ( s ) ψ ( s D ) + s = M + 1 M 1 T 2 2 C 2 ( s ) ψ ( s D ) + 2 s = 1 m h M 1 h T 1 T 2 C 12 ( s ) ψ ( s D + c Δ t )
c Δ t + s D = 0
( h + 1 ) λ 0 c Δ t ( m + h + M 1 ) λ 0
B 1 ( r ) = 0 { r > m } and B 2 ( r ) = 0 { r < m + h + 1 }
r max = m + h + M 1
Visibility ( OPD ) = s = r max r max 2 T 1 T 2 C 12 ( s ) · g ( [ s + S m ( OPD ) ] · λ 0 ) s = r max r max [ T 1 2 C 1 + T 2 2 C 2 ] · g ( s · λ 0 )

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