Abstract

Using a single optical fiber and miniature distal optics, spectrally-encoded endoscopy (SEE) has been demonstrated as a promising, three-dimensional endoscopic imaging method with a large number of resolvable points and high frame rates. We present a detailed theoretical study of the SEE prototype system and probe. Several key imaging parameters of SEE are thoroughly derived and formulated, including the three-dimensional point-spread function and field of view, as well as the system’s optical aberrations and fundamental limits. We find that the point-spread function of the SEE system maintains a unique relation between its transverse and axial shapes, discuss the asymmetry of the volumetric field of view, determine that the number of lateral resolvable points is nearly twice than what was previously accepted, and derive an expression for the upper limit for the total number of resolvable points in the cross-sectional image plane.

© 2009 Optical Society of America

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References

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  1. C. M. Brown, P. G. Reinhall, S. Karasawa, and E. J. Seibel, “Optomechanical design and fabrication of resonant microscanners for a scanning fiber endoscope,” Opt. Eng. 45, 043001-043010 (2006).
    [Crossref]
  2. D. L. Dickensheets and G. S. Kino, “Silicon-micromachined scanning confocal optical microscope,” J. Microelectromech. Syst. 7, 38–47 (1998).
    [Crossref]
  3. A. L. Polglase, W. J. McLaren, S. A. Skinner, R. Kiesslich, M. F. Neurath, and P. M. Delaney, “A fluorescence confocal endomicroscope for in vivo microscopy of the upper- and the lower-GI tract,” in Digestive Disease Week/105th Annual Meeting of the American-Gastroenterological-Association (New Orleans, LA, 2004), pp. 686–695.
  4. Y. C. Wu, Y. X. Leng, J. F. Xi, and X. D. Li, “Scanning all-fiber-optic endomicroscopy system for 3D nonlinear optical imaging of biological tissues,” Opt. Express 17(10), 7907–7915 (2009).
    [Crossref] [PubMed]
  5. G. J. Tearney, M. Shishkov, and B. E. Bouma, “Spectrally encoded miniature endoscopy,” Opt. Lett. 27(6), 412–414 (2002).
    [Crossref] [PubMed]
  6. D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
    [Crossref] [PubMed]
  7. D. Yelin, S. H. Yun, B. E. Bouma, and G. J. Tearney, “Three-dimensional imaging using spectral encoding heterodyne interferometry,” Opt. Lett. 30(14), 1794–1796 (2005).
    [Crossref] [PubMed]
  8. L. Froehly, S. N. Martin, T. Lasser, C. Depeursinge, and F. Lang, “Multiplexed 3D imaging using wavelength encoded spectral interferometry: a proof of principle,” Opt. Commun. 222, 127–136 (2003).
    [Crossref]
  9. D. Yelin, W. M. White, J. T. Motz, S. H. Yun, B. E. Bouma, and G. J. Tearney, “Spectral-domain spectrally-encoded endoscopy,” Opt. Express 15(5), 2432–2444 (2007).
    [Crossref] [PubMed]
  10. M. A. Choma, M. V. Sarunic, C. H. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (2003).
    [Crossref] [PubMed]
  11. J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003).
    [Crossref] [PubMed]
  12. R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003).
    [Crossref] [PubMed]
  13. D. Yelin, B. E. Bouma, N. Iftimia, and G. J. Tearney, “Three-dimensional spectrally encoded imaging,” Opt. Lett. 28(23), 2321–2323 (2003).
    [Crossref] [PubMed]
  14. D. Yelin, B. E. Bouma, and G. J. Tearney, “Volumetric sub-surface imaging using spectrally encoded endoscopy,” Opt. Express 16(3), 1748–1757 (2008).
    [Crossref] [PubMed]
  15. D. Yelin, B. E. Bouma, J. J. Rosowsky, and G. J. Tearney, “Doppler imaging using spectrally-encoded endoscopy,” Opt. Express 16(19), 14836–14844 (2008).
    [Crossref] [PubMed]
  16. B. E. Bouma, and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, New York, 2002).
  17. M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27(16), 1415–1417 (2002).
    [Crossref] [PubMed]
  18. K. B. Sung, C. N. Liang, M. Descour, T. Collier, M. Follen, and R. Richards-Kortum, “Fiber-optic confocal reflectance microscope with miniature objective for in vivo imaging of human tissues,” IEEE Trans. Biomed. Eng. 49(10), 1168–1172 (2002).
    [Crossref] [PubMed]
  19. C. Boudoux, S. Yun, W. Oh, W. White, N. Iftimia, M. Shishkov, B. Bouma, and G. Tearney, “Rapid wavelength-swept spectrally encoded confocal microscopy,” Opt. Express 13(20), 8214–8221 (2005).
    [Crossref] [PubMed]
  20. D. Yelin, C. Boudoux, B. E. Bouma, and G. J. Tearney, “Large area confocal microscopy,” Opt. Lett. 32(9), 1102–1104 (2007).
    [Crossref] [PubMed]
  21. D. Yelin, B. E. Bouma, S. H. Yun, and G. J. Tearney, “Double-clad fiber for endoscopy,” Opt. Lett. 29(20), 2408–2410 (2004).
    [Crossref] [PubMed]

2009 (1)

2008 (2)

2007 (2)

2006 (2)

C. M. Brown, P. G. Reinhall, S. Karasawa, and E. J. Seibel, “Optomechanical design and fabrication of resonant microscanners for a scanning fiber endoscope,” Opt. Eng. 45, 043001-043010 (2006).
[Crossref]

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

2005 (2)

2004 (1)

2003 (5)

2002 (3)

K. B. Sung, C. N. Liang, M. Descour, T. Collier, M. Follen, and R. Richards-Kortum, “Fiber-optic confocal reflectance microscope with miniature objective for in vivo imaging of human tissues,” IEEE Trans. Biomed. Eng. 49(10), 1168–1172 (2002).
[Crossref] [PubMed]

G. J. Tearney, M. Shishkov, and B. E. Bouma, “Spectrally encoded miniature endoscopy,” Opt. Lett. 27(6), 412–414 (2002).
[Crossref] [PubMed]

M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. 27(16), 1415–1417 (2002).
[Crossref] [PubMed]

1998 (1)

D. L. Dickensheets and G. S. Kino, “Silicon-micromachined scanning confocal optical microscope,” J. Microelectromech. Syst. 7, 38–47 (1998).
[Crossref]

Boudoux, C.

Bouma, B.

Bouma, B. E.

D. Yelin, B. E. Bouma, J. J. Rosowsky, and G. J. Tearney, “Doppler imaging using spectrally-encoded endoscopy,” Opt. Express 16(19), 14836–14844 (2008).
[Crossref] [PubMed]

D. Yelin, B. E. Bouma, and G. J. Tearney, “Volumetric sub-surface imaging using spectrally encoded endoscopy,” Opt. Express 16(3), 1748–1757 (2008).
[Crossref] [PubMed]

D. Yelin, C. Boudoux, B. E. Bouma, and G. J. Tearney, “Large area confocal microscopy,” Opt. Lett. 32(9), 1102–1104 (2007).
[Crossref] [PubMed]

D. Yelin, W. M. White, J. T. Motz, S. H. Yun, B. E. Bouma, and G. J. Tearney, “Spectral-domain spectrally-encoded endoscopy,” Opt. Express 15(5), 2432–2444 (2007).
[Crossref] [PubMed]

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

D. Yelin, S. H. Yun, B. E. Bouma, and G. J. Tearney, “Three-dimensional imaging using spectral encoding heterodyne interferometry,” Opt. Lett. 30(14), 1794–1796 (2005).
[Crossref] [PubMed]

D. Yelin, B. E. Bouma, S. H. Yun, and G. J. Tearney, “Double-clad fiber for endoscopy,” Opt. Lett. 29(20), 2408–2410 (2004).
[Crossref] [PubMed]

J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003).
[Crossref] [PubMed]

D. Yelin, B. E. Bouma, N. Iftimia, and G. J. Tearney, “Three-dimensional spectrally encoded imaging,” Opt. Lett. 28(23), 2321–2323 (2003).
[Crossref] [PubMed]

G. J. Tearney, M. Shishkov, and B. E. Bouma, “Spectrally encoded miniature endoscopy,” Opt. Lett. 27(6), 412–414 (2002).
[Crossref] [PubMed]

Brown, C. M.

C. M. Brown, P. G. Reinhall, S. Karasawa, and E. J. Seibel, “Optomechanical design and fabrication of resonant microscanners for a scanning fiber endoscope,” Opt. Eng. 45, 043001-043010 (2006).
[Crossref]

Cense, B.

Choma, M. A.

Collier, T.

K. B. Sung, C. N. Liang, M. Descour, T. Collier, M. Follen, and R. Richards-Kortum, “Fiber-optic confocal reflectance microscope with miniature objective for in vivo imaging of human tissues,” IEEE Trans. Biomed. Eng. 49(10), 1168–1172 (2002).
[Crossref] [PubMed]

de Boer, J. F.

Depeursinge, C.

L. Froehly, S. N. Martin, T. Lasser, C. Depeursinge, and F. Lang, “Multiplexed 3D imaging using wavelength encoded spectral interferometry: a proof of principle,” Opt. Commun. 222, 127–136 (2003).
[Crossref]

Descour, M.

K. B. Sung, C. N. Liang, M. Descour, T. Collier, M. Follen, and R. Richards-Kortum, “Fiber-optic confocal reflectance microscope with miniature objective for in vivo imaging of human tissues,” IEEE Trans. Biomed. Eng. 49(10), 1168–1172 (2002).
[Crossref] [PubMed]

Dickensheets, D. L.

D. L. Dickensheets and G. S. Kino, “Silicon-micromachined scanning confocal optical microscope,” J. Microelectromech. Syst. 7, 38–47 (1998).
[Crossref]

Fercher, A.

Fercher, A. F.

Follen, M.

K. B. Sung, C. N. Liang, M. Descour, T. Collier, M. Follen, and R. Richards-Kortum, “Fiber-optic confocal reflectance microscope with miniature objective for in vivo imaging of human tissues,” IEEE Trans. Biomed. Eng. 49(10), 1168–1172 (2002).
[Crossref] [PubMed]

Froehly, L.

L. Froehly, S. N. Martin, T. Lasser, C. Depeursinge, and F. Lang, “Multiplexed 3D imaging using wavelength encoded spectral interferometry: a proof of principle,” Opt. Commun. 222, 127–136 (2003).
[Crossref]

Hasan, T.

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

Hitzenberger, C.

Iftimia, N.

Izatt, J. A.

Karasawa, S.

C. M. Brown, P. G. Reinhall, S. Karasawa, and E. J. Seibel, “Optomechanical design and fabrication of resonant microscanners for a scanning fiber endoscope,” Opt. Eng. 45, 043001-043010 (2006).
[Crossref]

Kino, G. S.

D. L. Dickensheets and G. S. Kino, “Silicon-micromachined scanning confocal optical microscope,” J. Microelectromech. Syst. 7, 38–47 (1998).
[Crossref]

Kowalczyk, A.

Lang, F.

L. Froehly, S. N. Martin, T. Lasser, C. Depeursinge, and F. Lang, “Multiplexed 3D imaging using wavelength encoded spectral interferometry: a proof of principle,” Opt. Commun. 222, 127–136 (2003).
[Crossref]

Lasser, T.

L. Froehly, S. N. Martin, T. Lasser, C. Depeursinge, and F. Lang, “Multiplexed 3D imaging using wavelength encoded spectral interferometry: a proof of principle,” Opt. Commun. 222, 127–136 (2003).
[Crossref]

Leitgeb, R.

Leng, Y. X.

Li, X. D.

Liang, C. N.

K. B. Sung, C. N. Liang, M. Descour, T. Collier, M. Follen, and R. Richards-Kortum, “Fiber-optic confocal reflectance microscope with miniature objective for in vivo imaging of human tissues,” IEEE Trans. Biomed. Eng. 49(10), 1168–1172 (2002).
[Crossref] [PubMed]

Martin, S. N.

L. Froehly, S. N. Martin, T. Lasser, C. Depeursinge, and F. Lang, “Multiplexed 3D imaging using wavelength encoded spectral interferometry: a proof of principle,” Opt. Commun. 222, 127–136 (2003).
[Crossref]

Motz, J. T.

D. Yelin, W. M. White, J. T. Motz, S. H. Yun, B. E. Bouma, and G. J. Tearney, “Spectral-domain spectrally-encoded endoscopy,” Opt. Express 15(5), 2432–2444 (2007).
[Crossref] [PubMed]

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

Oh, W.

Park, B. H.

Pierce, M. C.

Reinhall, P. G.

C. M. Brown, P. G. Reinhall, S. Karasawa, and E. J. Seibel, “Optomechanical design and fabrication of resonant microscanners for a scanning fiber endoscope,” Opt. Eng. 45, 043001-043010 (2006).
[Crossref]

Richards-Kortum, R.

K. B. Sung, C. N. Liang, M. Descour, T. Collier, M. Follen, and R. Richards-Kortum, “Fiber-optic confocal reflectance microscope with miniature objective for in vivo imaging of human tissues,” IEEE Trans. Biomed. Eng. 49(10), 1168–1172 (2002).
[Crossref] [PubMed]

Rizvi, I.

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

Rosowsky, J. J.

Sarunic, M. V.

Seibel, E. J.

C. M. Brown, P. G. Reinhall, S. Karasawa, and E. J. Seibel, “Optomechanical design and fabrication of resonant microscanners for a scanning fiber endoscope,” Opt. Eng. 45, 043001-043010 (2006).
[Crossref]

Shishkov, M.

Sung, K. B.

K. B. Sung, C. N. Liang, M. Descour, T. Collier, M. Follen, and R. Richards-Kortum, “Fiber-optic confocal reflectance microscope with miniature objective for in vivo imaging of human tissues,” IEEE Trans. Biomed. Eng. 49(10), 1168–1172 (2002).
[Crossref] [PubMed]

Tearney, G.

Tearney, G. J.

D. Yelin, B. E. Bouma, J. J. Rosowsky, and G. J. Tearney, “Doppler imaging using spectrally-encoded endoscopy,” Opt. Express 16(19), 14836–14844 (2008).
[Crossref] [PubMed]

D. Yelin, B. E. Bouma, and G. J. Tearney, “Volumetric sub-surface imaging using spectrally encoded endoscopy,” Opt. Express 16(3), 1748–1757 (2008).
[Crossref] [PubMed]

D. Yelin, C. Boudoux, B. E. Bouma, and G. J. Tearney, “Large area confocal microscopy,” Opt. Lett. 32(9), 1102–1104 (2007).
[Crossref] [PubMed]

D. Yelin, W. M. White, J. T. Motz, S. H. Yun, B. E. Bouma, and G. J. Tearney, “Spectral-domain spectrally-encoded endoscopy,” Opt. Express 15(5), 2432–2444 (2007).
[Crossref] [PubMed]

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

D. Yelin, S. H. Yun, B. E. Bouma, and G. J. Tearney, “Three-dimensional imaging using spectral encoding heterodyne interferometry,” Opt. Lett. 30(14), 1794–1796 (2005).
[Crossref] [PubMed]

D. Yelin, B. E. Bouma, S. H. Yun, and G. J. Tearney, “Double-clad fiber for endoscopy,” Opt. Lett. 29(20), 2408–2410 (2004).
[Crossref] [PubMed]

J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003).
[Crossref] [PubMed]

D. Yelin, B. E. Bouma, N. Iftimia, and G. J. Tearney, “Three-dimensional spectrally encoded imaging,” Opt. Lett. 28(23), 2321–2323 (2003).
[Crossref] [PubMed]

G. J. Tearney, M. Shishkov, and B. E. Bouma, “Spectrally encoded miniature endoscopy,” Opt. Lett. 27(6), 412–414 (2002).
[Crossref] [PubMed]

White, W.

White, W. M.

D. Yelin, W. M. White, J. T. Motz, S. H. Yun, B. E. Bouma, and G. J. Tearney, “Spectral-domain spectrally-encoded endoscopy,” Opt. Express 15(5), 2432–2444 (2007).
[Crossref] [PubMed]

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

Wojtkowski, M.

Wu, Y. C.

Xi, J. F.

Yang, C. H.

Yelin, D.

Yun, S.

Yun, S. H.

IEEE Trans. Biomed. Eng. (1)

K. B. Sung, C. N. Liang, M. Descour, T. Collier, M. Follen, and R. Richards-Kortum, “Fiber-optic confocal reflectance microscope with miniature objective for in vivo imaging of human tissues,” IEEE Trans. Biomed. Eng. 49(10), 1168–1172 (2002).
[Crossref] [PubMed]

J. Microelectromech. Syst. (1)

D. L. Dickensheets and G. S. Kino, “Silicon-micromachined scanning confocal optical microscope,” J. Microelectromech. Syst. 7, 38–47 (1998).
[Crossref]

Nature (1)

D. Yelin, I. Rizvi, W. M. White, J. T. Motz, T. Hasan, B. E. Bouma, and G. J. Tearney, “Three-dimensional miniature endoscopy,” Nature 443(7113), 765 (2006).
[Crossref] [PubMed]

Opt. Commun. (1)

L. Froehly, S. N. Martin, T. Lasser, C. Depeursinge, and F. Lang, “Multiplexed 3D imaging using wavelength encoded spectral interferometry: a proof of principle,” Opt. Commun. 222, 127–136 (2003).
[Crossref]

Opt. Eng. (1)

C. M. Brown, P. G. Reinhall, S. Karasawa, and E. J. Seibel, “Optomechanical design and fabrication of resonant microscanners for a scanning fiber endoscope,” Opt. Eng. 45, 043001-043010 (2006).
[Crossref]

Opt. Express (7)

Opt. Lett. (7)

Other (2)

B. E. Bouma, and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, New York, 2002).

A. L. Polglase, W. J. McLaren, S. A. Skinner, R. Kiesslich, M. F. Neurath, and P. M. Delaney, “A fluorescence confocal endomicroscope for in vivo microscopy of the upper- and the lower-GI tract,” in Digestive Disease Week/105th Annual Meeting of the American-Gastroenterological-Association (New Orleans, LA, 2004), pp. 686–695.

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Figures (10)

Fig. 1
Fig. 1

Schematic of an interferometric spectral domain SEE system (ND – neutral density filter).

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Fig. 2
Fig. 2

(a) Schematic illustration of the SEE probe [6,14,15]. (b) A photograph of the probe.

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Fig. 3
Fig. 3

Schematic illustration of the illumination and signal collection optical paths in SEE.

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Fig. 4
Fig. 4

(a) Lateral intensity PSF (black solid line) and the spectral impulse response |h(λ)| (black dashed line) of SEE probe with a uniformly illuminated circular aperture and a working distance of 10 mm. A rectangular aperture produces a slightly different lateral spectral transfer function (hollow red squares) and PSF (filled red squares). (b) Interferometric axial PSF of confocal SEE (black solid) compared to the PSF of a confocal system with rectangular aperture (red dashed). The top axis axial coordinates assume ns = 1.

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Fig. 5
Fig. 5

Simulations of the light rays illuminating the sample through the SEE probe. A magnified view of the probe’s drawing is shown in the inset.

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Fig. 6
Fig. 6

Focal shift induced by chromatic aberrations for a working distance of 10 mm for the center wavelength.

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Fig. 7
Fig. 7

(a) Focal shift of different wavelengths, showing the effect of astigmatism. (b) Intensity FWHM – actual focus in x and y dimensions, represented with the diffraction-limited resolution for comparison.

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Fig. 8
Fig. 8

(a) Schematic of an SEE probe with 0.5 mm working distance ( λ = 800 nm). Note that the distal silica spacer is only 0.5 mm long. (b) The PSF in the x and y axes 0.5 mm from the grating. (c) The effect of working distance on the PSF in the x axis (green) and y axis (blue), resulting from the illuminating field curvature.

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Fig. 9
Fig. 9

The relative contribution of chromatic aberrations and astigmatism to the focal plane geometry.

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Fig. 10
Fig. 10

The three-dimensional field of view of SEE: (a) Volumetric rendering of the FOV, with the probe at the origin, rotated by total angle of 40°. (b) Surface rendering of the focal plane for a 40° rotation angle. (c) An x-z cross section of the FOV at y = 0. (d) FOV with full probe rotation (probe location is marked by a star). The x' and z' axes in (a) and (d) define a new coordinate system rotated by 2 θ 0 relative to the SEE focal plane.

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Equations (24)

Equations on this page are rendered with MathJax. Learn more.

n p sin θ 0 + n m sin θ = m λ G ,
δ x = G z 0 cos θ δ λ ≈ 2 π G z 0 cos θ δ k k 2 .
E M ( k m , x , y , z 0 ) = E 0 ( k m ) C k m 2 e i k m 2 z 0 ( ( x − x m ) 2 + y 2 ) [ J 1 ( k m D ( x − x m ) 2 + y 2 / 2 z 0 ) k m D ( x − x m ) 2 + y 2 / 2 z 0 ] ,
E C ( k , x s , z 0 ) = C E S ( k , x s , z 0 ) k 2 e i k 2 z 0 ( x s − x k ) 2 [ J 1 ( k D | x s − x k | / 2 z 0 ) k D | x s − x k | / 2 z 0 ] ,
E S ( k , x s , z 0 ) = r 0 ∫ σ k d k m E M ( k m , x s , z 0 ) f ( k m , k ) .
E C ( k , k s , z 0 ) = E 0 ( k ) r 0 C 2 k 4 e i k z 0 ( 2 π G z 0 cos θ 0 k − k s k 2 ) 2 [ J 1 ( π D G | k − k s | / k cos θ 0 ) π D G | k − k s | / k cos θ 0 ] 2 ,
h ( k , k s , z 0 ) = E C ( k , k s , z 0 ) / ( E 0 ( k ) ⋅ r 0 ) ,
Δ λ = 1.029 λ cos θ 0 D G .
S X = 0.738 λ z 0 D ,
E C ( k ) = E 0 ( k ) ∫ σ k ∫ D O F h ( k , k s ) r S ( x ( k s ) , z ) e 2 i k n ˜ z d z d k s ,
E R ( k ) = E 0 ( k ) r R e 2 i k z R ,
I ( k ) = | E R + E S | 2 = | E 0 ( k ) | 2 [ r R 2 + | ∫ σ k ∫ D O F h ( k , k s ) r S ( x ( k s ) , z ) e 2 i k n ˜ z d z d k s | 2                                                                                     + 2 r R Re { ∫ σ k ∫ D O F h ( k , k s ) r S ( x ( k s ) , z ) e 2 i k ( n ˜ z − z R ) d z d k s } ] .
I O C T ( k ) = | E 0 ( k ) | 2 ⋅ [ r R 2 + | ∫ D O F r S ( z ) e 2 i k n ˜ z d z | 2 + 2 r R ∫ D O F r S ( z ) cos ( 2 k ( n ˜ z − z R ) ) d z ] ,
I C I ( k ) = 2 r R E 0 2 ∫ σ k ∫ D O F h ( k , k s ) r S ( x ( k s ) , z ) cos ( k ζ ) d z d k s ,
I C I ( k ) = 2 r R r S ( 0 , z 0 ) E 0 2 h ( k − k 0 ) cos ( k ζ 0 ) ,
I ( ζ ) = I 0 ⋅ F T { I C I ( k ) } = I 0 ⋅ H ( ζ ) ⊗ [ δ ( ζ − ζ 0 ) + δ ( ζ + ζ 0 ) ] ,
S Z = 0.45 D G λ 0 n S cos θ 0 = 0.463 λ 0 2 n S Δ λ .
Δ X = G z 0 cos θ 0 σ λ .
Δ Y = 2 z 0 tan ( Δ φ ) ,
Δ Z = λ 2 Δ λ s p e c .
N X = Δ X S X = 1.355 σ λ G D λ 0 cos θ 0 .
N Y = Δ Y S Y = 1.355 2 D tan ( Δ φ ) λ 0 .
N Z = Δ λ / Δ λ s p e c .
N X N Z = 1.39 σ λ Δ λ s p e c .

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