Abstract

Straightforward numerical integration of the Rayleigh-Sommerfeld diffraction integral (R-SDI) remains computationally challenging, even with today’s computational resources. As such, approximating the R-SDI to decrease the computation time while maintaining a good accuracy is still a topic of interest. In this paper, we apply an approximation for the R-SDI that is to be used to propagate the field exiting a Coherent Fiber Bundle (CFB) with ultra-high numerical aperture (0.928) of which we presented the design and modal properties in previous work. Since our CFB has single-mode cores with a diameter (550nm) smaller than the wavelength (850nm) for which the CFB was designed, we approximate the highly divergent fundamental modes of the cores with real Dirac delta functions. We find that with this approximation we can strongly reduce the computation time of the R-SDI while maintaining a good agreement with the results of the full R-SDI. Using this approximation, we first determine the Point Spread Function (PSF) for an ‘ideal’ output field exiting the CFB (identical amplitudes for cores on a perfect hexagonal lattice with the phase of each core determined by the appropriate spherical and tilted plane wave front). Next, we analyze the PSF when amplitude or phase noise is superposed onto this ‘ideal’ field. We find that even in the presence of these types of noise, the effect on the central peak of PSF is limited. From these types of noise, phase noise is found to have the biggest impact on the PSF.

© 2013 Optical Society of America

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  1. C. M. Lee, C. J. Engelbrecht, T. D. Soper, F. Helmchen, and E. J. Seibel, “Scanning fiber endoscopy with highly flexible, 1 mm catheterscopes for wide-field, full-color imaging,” J Biophotonics 3(5-6), 385–407 (2010).
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  6. L. V. Doronina-Amitonova, I. V. Fedotov, A. B. Fedotov, and A. M. Zheltikov, “High-resolution wide-field Raman imaging through a fiber bundle,” Appl. Phys. Lett. 102(16), 161113 (2013).
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    [Crossref] [PubMed]
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2013 (2)

L. V. Doronina-Amitonova, I. V. Fedotov, A. B. Fedotov, and A. M. Zheltikov, “High-resolution wide-field Raman imaging through a fiber bundle,” Appl. Phys. Lett. 102(16), 161113 (2013).
[Crossref]

S. Heyvaert, H. Ottevaere, I. Kujawa, R. Buczynski, M. Raes, H. Terryn, and H. Thienpont, “Numerical characterization of an ultra-high NA coherent fiber bundle part I: modal analysis,” Opt. Express 21(19), 21991–22011 (2013).
[Crossref] [PubMed]

2012 (4)

J. P. Moore and M. D. Rogge, “Shape sensing using multi-core fiber optic cable and parametric curve solutions,” Opt. Express 20(3), 2967–2973 (2012).
[Crossref] [PubMed]

H. C. Park, C. Song, M. Kang, Y. Jeong, and K. H. Jeong, “Forward imaging OCT endoscopic catheter based on MEMS lens scanning,” Opt. Lett. 37(13), 2673–2675 (2012).
[Crossref] [PubMed]

S. Heyvaert, C. Debaes, H. Ottevaere, and H. Thienpont, “Design of a novel multicore optical fibre for imaging and beam delivery in endoscopy,” Proc. SPIE 8429, Optical Modelling and Design II, 84290Q, 84290Q-13 (2012).
[Crossref]

T. Cižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat Commun 3, 1027 (2012).
[Crossref] [PubMed]

2011 (3)

2010 (6)

J.-H. Han, J. Lee, and J. U. Kang, “Pixelation effect removal from fiber bundle probe based optical coherence tomography imaging,” Opt. Express 18(7), 7427–7439 (2010).
[Crossref] [PubMed]

A. Wuttig, M. Kanka, H. J. Kreuzer, and R. Riesenberg, “Packed domain Rayleigh-Sommerfeld wavefield propagation for large targets,” Opt. Express 18(26), 27036–27047 (2010).
[Crossref] [PubMed]

C. M. Lee, C. J. Engelbrecht, T. D. Soper, F. Helmchen, and E. J. Seibel, “Scanning fiber endoscopy with highly flexible, 1 mm catheterscopes for wide-field, full-color imaging,” J Biophotonics 3(5-6), 385–407 (2010).
[Crossref] [PubMed]

J. U. Kang, J.-H. Han, X. Liu, and K. Zhang, “Common-path optical coherence tomography for biomedical imaging and sensing,” J Opt Soc Korea 14(1), 1–13 (2010).
[Crossref] [PubMed]

M. Kyrish, R. Kester, R. Richards-Kortum, and T. Tkaczyk, “Improving spatial resolution of a fiber bundle optical biopsy,” Proc. SPIE 7558, Endoscopic Microscopy V, 755807, 755807-9 (2010).
[Crossref]

T. Cizmar and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4(6), 388–394 (2010).
[Crossref]

2009 (1)

W. Wang, K. Zhang, Q. Ren, and J. U. Kang, “Comparison of different focusing systems for common-path optical coherence tomography with fiber-optic bundle as endoscopic probe,” Opt. Eng.  48(10), 103001 (2009).

2008 (1)

D. Lorenc, M. Aranyosiova, R. Buczynski, R. Stepien, I. Bugar, A. Vincze, and D. Velic, “Nonlinear refractive index of multicomponent glasses designed for fabrication of photonic crystal fibers,” Appl. Phys. B 93(2–3), 531–538 (2008).
[Crossref]

2007 (1)

H. D. Ford and R. P. Tam, “Fibre imaging bundles for full-field optical coherence tomography,” Meas. Sci. Technol. 18(9), 2949–2957 (2007).
[Crossref]

2005 (1)

2000 (2)

M. J. Gander, D. Macrae, E. A. C. Galliot, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, and M. N. Inci, “Two-axis bend measurement using multicore optical fibre,” Opt. Commun. 182(1–3), 115–121 (2000).
[Crossref]

P. M. Lane, A. L. P. Dlugan, R. Richards-Kortum, and C. E. Macaulay, “Fiber-optic confocal microscopy using a spatial light modulator,” Opt. Lett. 25(24), 1780–1782 (2000).
[Crossref] [PubMed]

1984 (1)

L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” Geophys. J. Int. 79(1), 77–88 (1984).
[Crossref]

1983 (1)

1981 (1)

M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24(1), 355–359 (1981).
[Crossref]

1979 (1)

1977 (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Agrawal, G. P.

Aranyosiova, M.

D. Lorenc, M. Aranyosiova, R. Buczynski, R. Stepien, I. Bugar, A. Vincze, and D. Velic, “Nonlinear refractive index of multicomponent glasses designed for fabrication of photonic crystal fibers,” Appl. Phys. B 93(2–3), 531–538 (2008).
[Crossref]

Belanger, P. A.

M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24(1), 355–359 (1981).
[Crossref]

Bellanger, P.-A.

Bianchi, S.

Blanchard, P. M.

M. J. Gander, D. Macrae, E. A. C. Galliot, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, and M. N. Inci, “Two-axis bend measurement using multicore optical fibre,” Opt. Commun. 182(1–3), 115–121 (2000).
[Crossref]

Brenner, M.

Buczynski, R.

S. Heyvaert, H. Ottevaere, I. Kujawa, R. Buczynski, M. Raes, H. Terryn, and H. Thienpont, “Numerical characterization of an ultra-high NA coherent fiber bundle part I: modal analysis,” Opt. Express 21(19), 21991–22011 (2013).
[Crossref] [PubMed]

D. Lorenc, M. Aranyosiova, R. Buczynski, R. Stepien, I. Bugar, A. Vincze, and D. Velic, “Nonlinear refractive index of multicomponent glasses designed for fabrication of photonic crystal fibers,” Appl. Phys. B 93(2–3), 531–538 (2008).
[Crossref]

Bugar, I.

D. Lorenc, M. Aranyosiova, R. Buczynski, R. Stepien, I. Bugar, A. Vincze, and D. Velic, “Nonlinear refractive index of multicomponent glasses designed for fabrication of photonic crystal fibers,” Appl. Phys. B 93(2–3), 531–538 (2008).
[Crossref]

Burnett, J. G.

M. J. Gander, D. Macrae, E. A. C. Galliot, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, and M. N. Inci, “Two-axis bend measurement using multicore optical fibre,” Opt. Commun. 182(1–3), 115–121 (2000).
[Crossref]

Chen, Z.

Cizmar, T.

T. Cizmar and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4(6), 388–394 (2010).
[Crossref]

Cižmár, T.

T. Cižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat Commun 3, 1027 (2012).
[Crossref] [PubMed]

Couture, M.

P.-A. Bellanger and M. Couture, “Boundary diffraction of an inhomogeneous wave,” J. Opt. Soc. Am. 73(4), 446–450 (1983).
[Crossref]

M. Couture and P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24(1), 355–359 (1981).
[Crossref]

Debaes, C.

S. Heyvaert, C. Debaes, H. Ottevaere, and H. Thienpont, “Design of a novel multicore optical fibre for imaging and beam delivery in endoscopy,” Proc. SPIE 8429, Optical Modelling and Design II, 84290Q, 84290Q-13 (2012).
[Crossref]

Dholakia, K.

T. Cižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat Commun 3, 1027 (2012).
[Crossref] [PubMed]

T. Cizmar and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photonics 4(6), 388–394 (2010).
[Crossref]

Di Leonardo, R.

Dlugan, A. L. P.

Doronina-Amitonova, L. V.

L. V. Doronina-Amitonova, I. V. Fedotov, A. B. Fedotov, and A. M. Zheltikov, “High-resolution wide-field Raman imaging through a fiber bundle,” Appl. Phys. Lett. 102(16), 161113 (2013).
[Crossref]

Dunsby, C.

Elahi, S. F.

S. F. Elahi and T. D. Wang, “Future and advances in endoscopy,” J Biophotonics 4(7-8), 471–481 (2011).
[Crossref] [PubMed]

Engelbrecht, C. J.

C. M. Lee, C. J. Engelbrecht, T. D. Soper, F. Helmchen, and E. J. Seibel, “Scanning fiber endoscopy with highly flexible, 1 mm catheterscopes for wide-field, full-color imaging,” J Biophotonics 3(5-6), 385–407 (2010).
[Crossref] [PubMed]

Fedotov, A. B.

L. V. Doronina-Amitonova, I. V. Fedotov, A. B. Fedotov, and A. M. Zheltikov, “High-resolution wide-field Raman imaging through a fiber bundle,” Appl. Phys. Lett. 102(16), 161113 (2013).
[Crossref]

Fedotov, I. V.

L. V. Doronina-Amitonova, I. V. Fedotov, A. B. Fedotov, and A. M. Zheltikov, “High-resolution wide-field Raman imaging through a fiber bundle,” Appl. Phys. Lett. 102(16), 161113 (2013).
[Crossref]

Felsen, L. B.

L. B. Felsen, “Geometrical theory of diffraction, evanescent waves, complex rays and Gaussian beams,” Geophys. J. Int. 79(1), 77–88 (1984).
[Crossref]

Ford, H. D.

H. D. Ford and R. P. Tam, “Fibre imaging bundles for full-field optical coherence tomography,” Meas. Sci. Technol. 18(9), 2949–2957 (2007).
[Crossref]

French, P. M. W.

Galliot, E. A. C.

M. J. Gander, D. Macrae, E. A. C. Galliot, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, and M. N. Inci, “Two-axis bend measurement using multicore optical fibre,” Opt. Commun. 182(1–3), 115–121 (2000).
[Crossref]

Gander, M. J.

M. J. Gander, D. Macrae, E. A. C. Galliot, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, and M. N. Inci, “Two-axis bend measurement using multicore optical fibre,” Opt. Commun. 182(1–3), 115–121 (2000).
[Crossref]

Greenaway, A. H.

M. J. Gander, D. Macrae, E. A. C. Galliot, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, and M. N. Inci, “Two-axis bend measurement using multicore optical fibre,” Opt. Commun. 182(1–3), 115–121 (2000).
[Crossref]

Guo, S.

Han, J.-H.

J.-H. Han, J. Lee, and J. U. Kang, “Pixelation effect removal from fiber bundle probe based optical coherence tomography imaging,” Opt. Express 18(7), 7427–7439 (2010).
[Crossref] [PubMed]

J. U. Kang, J.-H. Han, X. Liu, and K. Zhang, “Common-path optical coherence tomography for biomedical imaging and sensing,” J Opt Soc Korea 14(1), 1–13 (2010).
[Crossref] [PubMed]

Helmchen, F.

C. M. Lee, C. J. Engelbrecht, T. D. Soper, F. Helmchen, and E. J. Seibel, “Scanning fiber endoscopy with highly flexible, 1 mm catheterscopes for wide-field, full-color imaging,” J Biophotonics 3(5-6), 385–407 (2010).
[Crossref] [PubMed]

Heyvaert, S.

S. Heyvaert, H. Ottevaere, I. Kujawa, R. Buczynski, M. Raes, H. Terryn, and H. Thienpont, “Numerical characterization of an ultra-high NA coherent fiber bundle part I: modal analysis,” Opt. Express 21(19), 21991–22011 (2013).
[Crossref] [PubMed]

S. Heyvaert, C. Debaes, H. Ottevaere, and H. Thienpont, “Design of a novel multicore optical fibre for imaging and beam delivery in endoscopy,” Proc. SPIE 8429, Optical Modelling and Design II, 84290Q, 84290Q-13 (2012).
[Crossref]

Inci, M. N.

M. J. Gander, D. Macrae, E. A. C. Galliot, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, and M. N. Inci, “Two-axis bend measurement using multicore optical fibre,” Opt. Commun. 182(1–3), 115–121 (2000).
[Crossref]

Jeong, K. H.

Jeong, Y.

Jones, J. D. C.

M. J. Gander, D. Macrae, E. A. C. Galliot, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, and M. N. Inci, “Two-axis bend measurement using multicore optical fibre,” Opt. Commun. 182(1–3), 115–121 (2000).
[Crossref]

Kang, J. U.

J.-H. Han, J. Lee, and J. U. Kang, “Pixelation effect removal from fiber bundle probe based optical coherence tomography imaging,” Opt. Express 18(7), 7427–7439 (2010).
[Crossref] [PubMed]

J. U. Kang, J.-H. Han, X. Liu, and K. Zhang, “Common-path optical coherence tomography for biomedical imaging and sensing,” J Opt Soc Korea 14(1), 1–13 (2010).
[Crossref] [PubMed]

W. Wang, K. Zhang, Q. Ren, and J. U. Kang, “Comparison of different focusing systems for common-path optical coherence tomography with fiber-optic bundle as endoscopic probe,” Opt. Eng.  48(10), 103001 (2009).

Kang, M.

Kanka, M.

Kester, R.

M. Kyrish, R. Kester, R. Richards-Kortum, and T. Tkaczyk, “Improving spatial resolution of a fiber bundle optical biopsy,” Proc. SPIE 7558, Endoscopic Microscopy V, 755807, 755807-9 (2010).
[Crossref]

Kreuzer, H. J.

Kujawa, I.

Kyrish, M.

M. Kyrish, R. Kester, R. Richards-Kortum, and T. Tkaczyk, “Improving spatial resolution of a fiber bundle optical biopsy,” Proc. SPIE 7558, Endoscopic Microscopy V, 755807, 755807-9 (2010).
[Crossref]

Lane, P. M.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Lee, C. M.

C. M. Lee, C. J. Engelbrecht, T. D. Soper, F. Helmchen, and E. J. Seibel, “Scanning fiber endoscopy with highly flexible, 1 mm catheterscopes for wide-field, full-color imaging,” J Biophotonics 3(5-6), 385–407 (2010).
[Crossref] [PubMed]

Lee, J.

Liu, X.

J. U. Kang, J.-H. Han, X. Liu, and K. Zhang, “Common-path optical coherence tomography for biomedical imaging and sensing,” J Opt Soc Korea 14(1), 1–13 (2010).
[Crossref] [PubMed]

Lorenc, D.

D. Lorenc, M. Aranyosiova, R. Buczynski, R. Stepien, I. Bugar, A. Vincze, and D. Velic, “Nonlinear refractive index of multicomponent glasses designed for fabrication of photonic crystal fibers,” Appl. Phys. B 93(2–3), 531–538 (2008).
[Crossref]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Macaulay, C. E.

Macrae, D.

M. J. Gander, D. Macrae, E. A. C. Galliot, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, and M. N. Inci, “Two-axis bend measurement using multicore optical fibre,” Opt. Commun. 182(1–3), 115–121 (2000).
[Crossref]

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

McBride, R.

M. J. Gander, D. Macrae, E. A. C. Galliot, R. McBride, J. D. C. Jones, P. M. Blanchard, J. G. Burnett, A. H. Greenaway, and M. N. Inci, “Two-axis bend measurement using multicore optical fibre,” Opt. Commun. 182(1–3), 115–121 (2000).
[Crossref]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial optics,” Phys. Rev. A 11(4), 1365–1370 (1975).
[Crossref]

Moore, J. P.

Mukai, D.

Neil, M. A. A.

Ottevaere, H.

S. Heyvaert, H. Ottevaere, I. Kujawa, R. Buczynski, M. Raes, H. Terryn, and H. Thienpont, “Numerical characterization of an ultra-high NA coherent fiber bundle part I: modal analysis,” Opt. Express 21(19), 21991–22011 (2013).
[Crossref] [PubMed]

S. Heyvaert, C. Debaes, H. Ottevaere, and H. Thienpont, “Design of a novel multicore optical fibre for imaging and beam delivery in endoscopy,” Proc. SPIE 8429, Optical Modelling and Design II, 84290Q, 84290Q-13 (2012).
[Crossref]

Park, H. C.

Paterson, C.

Pattanayak, D. N.

Raes, M.

Ren, Q.

W. Wang, K. Zhang, Q. Ren, and J. U. Kang, “Comparison of different focusing systems for common-path optical coherence tomography with fiber-optic bundle as endoscopic probe,” Opt. Eng.  48(10), 103001 (2009).

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[Crossref] [PubMed]

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Soper, T. D.

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M. Kyrish, R. Kester, R. Richards-Kortum, and T. Tkaczyk, “Improving spatial resolution of a fiber bundle optical biopsy,” Proc. SPIE 7558, Endoscopic Microscopy V, 755807, 755807-9 (2010).
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J. U. Kang, J.-H. Han, X. Liu, and K. Zhang, “Common-path optical coherence tomography for biomedical imaging and sensing,” J Opt Soc Korea 14(1), 1–13 (2010).
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Optical Modelling and Design (1)

S. Heyvaert, C. Debaes, H. Ottevaere, and H. Thienpont, “Design of a novel multicore optical fibre for imaging and beam delivery in endoscopy,” Proc. SPIE 8429, Optical Modelling and Design II, 84290Q, 84290Q-13 (2012).
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J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 3.

Schott website: http://www.schott.com/advanced_optics/english/abbe_datasheets/schott_datasheet_sf6.pdf?highlighted_text=SF6

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Figures (14)

Fig. 1
Fig. 1 Amplitude (a) of the field to be propagated and phase given to the fundamental mode of the cores to focus and steer the light exiting the CFB under angles θ = 0°, θ = 15° and θ = 30° ((b) through (d) respectively). The black circles in the xy-plane in (b)-(d) represent the position of the cores and are for clarification purposes only.

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Fig. 2
Fig. 2 Cross sections in the image plane along x-axis and y-axis (left and right column respectively) of the propagated field using the exact R-SDI and the approximation for θ = 0° (top row), θ = 15° (middle row) and θ = 30° (bottom row) show good agreement.

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Fig. 3
Fig. 3 Amplitude of the field (with variations in core size clearly visible) to be propagated with the standard R-SDI.

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Fig. 4
Fig. 4 Even for cores with different diameters, the cross sections of the propagated field using the exact R-SDI and the approximation show good agreement.

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Fig. 5
Fig. 5 The RMS error for the case with variable core sizes is higher than for the case with identical cores though the difference in RMS error between the two cases decreases for increasing θ.

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Fig. 6
Fig. 6 The different probability density functions used to simulate noise on the amplitude of the cores. Negative amplitudes were set to zero.

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Fig. 7
Fig. 7 The PSF at different viewing angles θ, without amplitude noise (left column) and with noise (middle and right column). The PSF for more values of θ and σΑ were calculated but are not shown here for the sake of conciseness.

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Fig. 8
Fig. 8 The cross sections through the focus along the x-axis (left column) and the y-axis (right column) vary little for varying σA even at large values of θ.

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Fig. 9
Fig. 9 The RMS error as function of σA for different values of θ.

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Fig. 10
Fig. 10 The PSF at different viewing angles θ, without (left column) and with phase noise (middle and right column). For σϕ >2.31 (column on the right), the resulting ‘PSF’ is just speckle.

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Fig. 11
Fig. 11 The curves for the RMS error as function of σϕ for different values of θ. For >2.31 the RMS error approximately remains the same independent of θ.

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Fig. 12
Fig. 12 The cross sections through the focus along the x-axis (left column) and the y-axis (right column) vary little for σϕ ≤0.11 even at large values of θ.

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Fig. 13
Fig. 13 The cross sections through the focus along the x-axis (left column) and the y-axis (right column) show discernible central peaks even at large values of θ for σϕ up to π/2.

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Fig. 14
Fig. 14 The cross sections through the focus along the x-axis (left column) and the y-axis (right column) resulting from the propagation of a field containing both amplitude and phase noise.

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Tables (1)

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Table 1 Design parameters for a CFB with scanning and focusing capabilities (from [15])

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Equations (6)

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δ( r → −( r → c −j z R 1 → b ) )
δ( x o − x c , y o − y c ,z+j z R )
U c r ( P o )=A e j φ r δ( x o − x c r , y o − y c r ) with r=1... N c
U( P i )=− 1 2π ∬ S U( P o )(jk− 1 R ) e jkR R z R d x o d y o with R= ( x o − x i ) 2 + ( y o − y i ) 2 + z 2
U c r ( P i )=− A e j φ r 2π (jk− 1 R c r ) e jk R c r R c r z R c r with R c r = ( x o − x c r ) 2 + ( y o − y c r ) 2 + z 2 .
1 N i ∑ q=1 N i | E q exact − E q approximation | 2

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