Abstract

Longitudinal spatial coherence (LSC) is determined by the spatial frequency content of an optical beam. The use of lenses with a high numerical aperture (NA) in full-field optical coherence tomography and a narrowband light source makes the LSC length much shorter than the temporal coherence length, hence suggesting that high-resolution 3D images of biological and multilayered samples can be obtained based on the low LSC. A simplified model is derived, supported by experimental results, which describes the expected interference output signal of multilayered samples when high-NA lenses are used together with a narrowband light source. An expression for the correction factor for the layer thickness determination is found valid for high-NA objectives. Additionally, the method was applied to a strongly scattering layer, demonstrating the potential of this method for high-resolution imaging of scattering media.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. 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]
  3. E. Beaurepaire, A. C. Boccara, M. Lebec, L. Blanchot, and H. Saint-Jalmes, “Full-field optical coherence microscopy,” Opt. Lett. 23, 244–246 (1998).
    [CrossRef]
  4. W.Krug, J.Rienitz, and G.Schultz, eds., Contributions to Interference Microscopy (Hilger & Watts, 1964).
  5. M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 775, 233–247 (1987).
  6. D. Gale, M. I. Pether, and J. C. Dainty, “Linnik microscope imaging of integrated circuit structures,” Appl. Opt. 35, 131–148 (1996).
    [CrossRef] [PubMed]
  7. L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. 27, 530–532 (2002).
    [CrossRef]
  8. A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, “High-resolution full-field optical coherence tomography with a Linnik microscope,” Appl. Opt. 41, 805–812 (2002).
    [CrossRef] [PubMed]
  9. B. Laude, A. De Martino, B. Drévillon, L. Benattar, and L. Schwartz, “Full-field optical coherence tomography with thermal light,” Appl. Opt. 41, 6637–6645 (2002).
    [CrossRef] [PubMed]
  10. M. Akiba, K. P. Chan, and N. Tanno, “Full-field optical coherence tomography by two-dimensional heterodyne detection with a pair of CCD cameras,” Opt. Lett. 28, 816–818(2003).
    [CrossRef] [PubMed]
  11. G. Moneron, A. C. Boccara, and A. Dubois, “Stroboscopic ultrahigh-resolution full-field optical coherence tomography,” Opt. Lett. 30, 1351–1353 (2005).
    [CrossRef] [PubMed]
  12. Y. Watanabe, Y. Hayasaka, M. Sato, and N. Tanno, “Full-field optical coherence tomography by achromatic phase shifting with a rotating polarizer,” Appl. Opt. 44, 1387–1392 (2005).
    [CrossRef] [PubMed]
  13. W. Y. Oh, B. E. Bouma, N. Iftimia, S. H. Yun, R. Yelin, and G. J. Tearney, “Ultrahigh-resolution full-field optical coherence microscopy using InGaAs camera,” Opt. Express 14, 726–735(2006).
    [CrossRef] [PubMed]
  14. M. S. Hrebesh, R. Dabu, and M. Sato, “In vivo imaging of dynamic biological specimen by real-time single-shot full-field optical coherence tomography,” Opt. Commun. 282, 674–683(2009).
    [CrossRef]
  15. P. de Groot and X. Colonna de Lega, “Signal modeling for low-coherence height-scanning interference microscopy,” Appl. Opt. 43, 4821–4830 (2004).
    [CrossRef] [PubMed]
  16. A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, “Ultrahigh-resolution full-field optical coherence tomography,” Appl. Opt. 43, 2874–2883 (2004).
    [CrossRef] [PubMed]
  17. L. Yu and M. K. Kim, “Full-color three-dimensional microscopy by wide-field optical coherence tomography,” Opt. Express 12, 6632–6641 (2004).
    [CrossRef] [PubMed]
  18. I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and interference microscopy,” J. Opt. A: Pure Appl. Opt. 8, 952–958(2006).
    [CrossRef]
  19. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39, 4107–4111(2000).
    [CrossRef]
  20. V. Ryabukho, D. Lyakin, and M. Lobachev, “Longitudinal pure spatial coherence of a light field with wide frequency and angular spectra,” Opt. Lett. 30, 224–226 (2005).
    [CrossRef] [PubMed]
  21. S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
    [CrossRef]
  22. S. H. Wiersma and T. D. Visser, “Defocusing of converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. 13, 320–325 (1996).
    [CrossRef]
  23. G. S. Kino and S. S. C. Chim, “Mirau correlation microscope,” Appl. Opt. 29, 3775–3783 (1990).
    [CrossRef] [PubMed]
  24. A. Dubois, G. Moneron, and C. Boccara, “Thermal-light full-field optical coherence tomography in the 1.2 μm wavelength region,” Opt. Commun. 266, 738–743 (2006).
    [CrossRef]
  25. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 77–78 and 157–158.

2009

M. S. Hrebesh, R. Dabu, and M. Sato, “In vivo imaging of dynamic biological specimen by real-time single-shot full-field optical coherence tomography,” Opt. Commun. 282, 674–683(2009).
[CrossRef]

2006

I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and interference microscopy,” J. Opt. A: Pure Appl. Opt. 8, 952–958(2006).
[CrossRef]

A. Dubois, G. Moneron, and C. Boccara, “Thermal-light full-field optical coherence tomography in the 1.2 μm wavelength region,” Opt. Commun. 266, 738–743 (2006).
[CrossRef]

W. Y. Oh, B. E. Bouma, N. Iftimia, S. H. Yun, R. Yelin, and G. J. Tearney, “Ultrahigh-resolution full-field optical coherence microscopy using InGaAs camera,” Opt. Express 14, 726–735(2006).
[CrossRef] [PubMed]

2005

2004

2003

2002

2000

1998

1996

S. H. Wiersma and T. D. Visser, “Defocusing of converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. 13, 320–325 (1996).
[CrossRef]

D. Gale, M. I. Pether, and J. C. Dainty, “Linnik microscope imaging of integrated circuit structures,” Appl. Opt. 35, 131–148 (1996).
[CrossRef] [PubMed]

1993

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[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]

1990

1987

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 775, 233–247 (1987).

Abdulhalim, I.

I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and interference microscopy,” J. Opt. A: Pure Appl. Opt. 8, 952–958(2006).
[CrossRef]

Akiba, M.

Beaurepaire, E.

Benattar, L.

Blanchot, L.

Boccara, A. C.

Boccara, C.

A. Dubois, G. Moneron, and C. Boccara, “Thermal-light full-field optical coherence tomography in the 1.2 μm wavelength region,” Opt. Commun. 266, 738–743 (2006).
[CrossRef]

A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L. Vabre, and C. Boccara, “Ultrahigh-resolution full-field optical coherence tomography,” Appl. Opt. 43, 2874–2883 (2004).
[CrossRef] [PubMed]

Bouma, B. E.

Chan, K. P.

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]

Chim, S. S. C.

Cohen, F.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 775, 233–247 (1987).

Colonna de Lega, X.

Cremer, C.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Dabu, R.

M. S. Hrebesh, R. Dabu, and M. Sato, “In vivo imaging of dynamic biological specimen by real-time single-shot full-field optical coherence tomography,” Opt. Commun. 282, 674–683(2009).
[CrossRef]

Dainty, J. C.

Davidson, M.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 775, 233–247 (1987).

de Groot, P.

De Martino, A.

Drévillon, B.

Dubois, A.

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.

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]

Gale, D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 77–78 and 157–158.

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]

Grieve, K.

Hayasaka, Y.

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]

Hell, S.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Hrebesh, M. S.

M. S. Hrebesh, R. Dabu, and M. Sato, “In vivo imaging of dynamic biological specimen by real-time single-shot full-field optical coherence tomography,” Opt. Commun. 282, 674–683(2009).
[CrossRef]

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]

Iftimia, N.

Kaufman, K.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 775, 233–247 (1987).

Kim, M. K.

Kino, G. S.

Laude, B.

Lebec, M.

Lecaque, R.

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]

Lobachev, M.

Lyakin, D.

Mazor, I.

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 775, 233–247 (1987).

Moneron, G.

Oh, W. Y.

Pether, M. I.

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]

Reiner, G.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

Rosen, J.

Ryabukho, V.

Saint-Jalmes, H.

Sato, M.

M. S. Hrebesh, R. Dabu, and M. Sato, “In vivo imaging of dynamic biological specimen by real-time single-shot full-field optical coherence tomography,” Opt. Commun. 282, 674–683(2009).
[CrossRef]

Y. Watanabe, Y. Hayasaka, M. Sato, and N. Tanno, “Full-field optical coherence tomography by achromatic phase shifting with a rotating polarizer,” Appl. Opt. 44, 1387–1392 (2005).
[CrossRef] [PubMed]

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]

Schwartz, L.

Stelzer, E. H. K.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

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.

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]

Takeda, M.

Tanno, N.

Tearney, G. J.

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]

Vabre, L.

Visser, T. D.

S. H. Wiersma and T. D. Visser, “Defocusing of converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. 13, 320–325 (1996).
[CrossRef]

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]

Watanabe, Y.

Wiersma, S. H.

S. H. Wiersma and T. D. Visser, “Defocusing of converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. 13, 320–325 (1996).
[CrossRef]

Yelin, R.

Yu, L.

Yun, S. H.

Appl. Opt.

J. Microsc.

S. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fluorescence microscopy induced by mismatches in refractive index,” J. Microsc. 169, 391–405 (1993).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

I. Abdulhalim, “Competence between spatial and temporal coherence in full field optical coherence tomography and interference microscopy,” J. Opt. A: Pure Appl. Opt. 8, 952–958(2006).
[CrossRef]

J. Opt. Soc. Am.

S. H. Wiersma and T. D. Visser, “Defocusing of converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. 13, 320–325 (1996).
[CrossRef]

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]

Opt. Commun.

M. S. Hrebesh, R. Dabu, and M. Sato, “In vivo imaging of dynamic biological specimen by real-time single-shot full-field optical coherence tomography,” Opt. Commun. 282, 674–683(2009).
[CrossRef]

A. Dubois, G. Moneron, and C. Boccara, “Thermal-light full-field optical coherence tomography in the 1.2 μm wavelength region,” Opt. Commun. 266, 738–743 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

M. Davidson, K. Kaufman, I. Mazor, and F. Cohen, “An application of interference microscopy to integrated circuit inspection and metrology,” Proc. SPIE 775, 233–247 (1987).

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

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 77–78 and 157–158.

W.Krug, J.Rienitz, and G.Schultz, eds., Contributions to Interference Microscopy (Hilger & Watts, 1964).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Interference microscopy optical system: LS, light source; OF, optical fiber; D, diffuser; BPF, bandpass filter; AS, aperture stop; FS, field stop; L, lens; BS, beam splitter; O ob and O r , objectives; PPs and PPr, parallel planes; S, sample; S M , sample mirror; R M , reference mirror.

Fig. 2
Fig. 2

Experimental interferogram obtained from a,  4.8 μm SiO 2 thin film on Si substrate and b, multilayer sample, from left to right, 6 μm Mylar with a refractive index of n Mylar = 1.6 , unknown width of free air and 4.8 μm of SiO 2 on Si substrate with a refractive index of n SiO 2 = 1.54 .

Fig. 3
Fig. 3

Comparison of simulated and experimental results from a 4.8 μm SiO 2 thin film on an Si substrate using an objective lens with two different effective NAs and a narrowband light source with the center wavelength at 580 nm and a FWHM of 10 nm . a, Simulation using Eq. 15 with NA = 0.65 . b, Experimental result using effective NA of 0.65 . c, Simulation using Eq. 15 with NA = 0.75 . d, Experimental result using an effective NA of 0.75 .

Fig. 4
Fig. 4

Experimental sample and a resolution target. a, Mirror with a scattering layer composed of silica spheres having a diameter of 1.86 μm mixed with UV glue. The image is grabbed with a digital camera showing the whole mirror of 1 diameter and the reflected image of the camera. b, En-face image of the sample grabbed through the scanning process showing a large number of the scattering silica spheres as white stains. c, Cross-sectional image of the scattering layer. d,  200 lines / mm Ronchi ruling resolution target.

Equations (19)

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

I D ( x , y ) | E image ( x , y ) | 2 | Δ ( cos θ ) e j ( 2 k n 0 cos θ · z S + ϕ k , S + 2 k L S ) H ( x , y , θ ) d ( cos θ ) + r R Δ ( cos θ ) e j ( 2 k n 0 cos θ · z R + ϕ k , R + 2 k L R ) d ( cos θ ) + c.c. | 2 .
H ( x , y , θ ) = 0 L r S ( x , y , z ) e j 2 k cos φ ( n + j κ ) z d z .
r S ( x , y , z ) = n i + 1 n i n i + 1 + n i δ ( z l ( x , y ) ) .
l ( x , y ) = { l ( x , y ) 0 < z 0 else .
cos φ 1 ( n 0 / n ) 2 ( 1 cos θ ) .
I D ( x , y ) Δ ( cos θ ) | H ( x , y , θ ) | 2 d ( cos θ ) + Δ ( cos θ ) | r R | 2 d ( cos θ ) + Δ ( cos θ ) 2 r R Re { H ( x , y , θ ) · e j [ 2 k n 0 cos θ · ( z R z S ) + ( ϕ k , R ϕ k , S ) + 2 k ( L R L S ) ] } d ( cos θ ) .
i 1 = NA 2 2 0 L ( n i + 1 n i n i + 1 + n i ) 2 δ ( z l ( x , y ) ) d z = NA 2 2 0 L R S ( x , y , z ) d z .
i 2 = Δ ( cos θ ) | r R | 2 d ( cos θ ) = R R Δ ( cos θ ) d ( cos θ ) = NA 2 2 R R ,
i 3 = Δ cos θ 2 r R Re { H ( x , y , θ ) · e j 2 k n 0 cos θ Δ z } d ( cos θ ) = 2 R R Δ cos θ 0 L r S ( z ) cos { 2 k cos θ [ ( n 0 / n ) 2 n l ( x , y ) n 0 Δ z ] + 2 k [ 1 ( n 0 / n ) 2 ] n l ( x , y ) } d z d ( cos θ ) = 2 R R NA 2 2 0 L r S ( z ) d z · sin c { 2 π λ [ n 0 Δ z ( n 0 / n ) 2 · n l ( x , y ) ] NA 2 2 } × cos { 2 π λ / 2 ( 1 NA 2 4 ) [ n 0 Δ z ( n 0 / n ) 2 · n l ( x , y ) ] 2 π λ / 2 [ 1 ( n 0 / n ) 2 ] n l ( x , y ) } .
I N ( x , y , Δ z ) = 1 + 2 R R 0 L R s ( x , y , z ) d z + R R 0 L r S ( x , y , z ) d z × sin c { 2 π λ [ Δ z n 0 l ( x , y ) n · ( n 0 / n ) 2 ] NA 2 2 } × cos ( 2 π λ / 2 ( 1 NA 2 / 4 ) { Δ z n 0 l ( x , y ) n · [ 1 ( n 0 / n ) 2 NA 2 / 4 ] ( 1 NA 2 / 4 ) } ) .
Δ z ( x , y ) peak N = i = 0 N l i ( x , y ) ( n 0 / n i ) .
I N ( x , y , Δ z ) 1 + 2 R R R scat ( x , y ) + 0 L R s ( x , y , z ) exp [ 2 α l ( x , y ) ] d z + R R × 0 L r S ( x , y , z ) exp [ α l ( x , y ) ] d z × sin c { 2 π λ [ Δ z n 0 l ( x , y ) n · ( n 0 / n ) 2 ] NA eff 2 2 } × cos ( 2 π λ / 2 ( 1 NA eff 2 / 4 ) { Δ z n 0 l ( x , y ) n · [ 1 ( n 0 / n ) 2 NA eff 2 / 4 ] ( 1 NA eff 2 / 4 ) } ) .
R Axial i = λ n i NA obj 2 .
i 1 = Δ ( cos θ ) | H ( x , y , θ ) | 2 d ( cos θ ) = Δ ( cos θ ) | 0 L r S ( x , y , z ) e j 2 k cos φ · n l ( x , y , z ) d z | 2 d ( cos θ ) = Δ ( cos θ ) 0 L 0 L r S ( x , y , z ) r S * ( x , y , z ) e j 2 k [ cos φ n l ( x , y , z ) cos φ n l ( x , y , z ) ] d z d z d ( cos θ ) = 0 L 0 L r S ( x , y , z ) r S * ( x , y , z ) d z d z Δ ( cos θ ) e j 2 k [ cos φ n l ( x , y , z ) cos φ n l ( x , y , z ) ] d ( cos θ ) .
i 1 = 0 L 0 L r S ( x , y , z ) r S * ( x , y , z ) d z d z e j 2 k { [ 1 ( n 0 / n ) 2 ] n [ l ( x , y , z ) l ( x , y , z ) ] } × Δ ( cos θ ) e j 2 k cos θ ( n 0 / n ) 2 n [ l ( x , y , z ) l ( x , y , z ) ] d ( cos θ ) .
i 1 = 0 L 0 L r S ( x , y , z ) r S * ( x , y , z ) d z d z e j 2 k { [ 1 ( n 0 / n ) 2 ] n [ l ( x , y , z ) l ( x , y , z ) ] } × e j 2 k cos θ max ( n 0 / n ) 2 n [ l ( x , y , z ) l ( x , y , z ) ] e j 2 k ( n 0 / n ) 2 n [ l ( x , y , z ) l ( x , y , z ) ] j 2 k ( n 0 / n ) 2 n [ l ( x , y , z ) l ( x , y , z ) ] .
e j α e j β = 2 sin ( α β 2 ) [ sin ( α + β 2 ) j cos ( α + β 2 ) ] ,
i 1 = NA 2 2 0 L 0 L r S ( x , y , z ) r S * ( x , y , z ) d z d z e j 2 k { [ 1 ( n 0 / n ) 2 ] n [ l ( x , y , z ) l ( x , y , z ) ] } × 1 j sin c { k NA 2 / 2 · ( n 0 / n ) 2 n [ l ( x , y , z ) l ( x , y , z ) ] } × ( sin { 2 k ( 1 NA 2 / 4 ) ( n 0 / n ) 2 n [ l ( x , y , z ) l ( x , y , z ) ] j cos { 2 k ( 1 NA 2 / 4 ) ( n 0 / n ) 2 n [ l ( x , y , z ) l ( x , y , z ) ] ) .
i 1 = NA 2 2 0 L ( n i + 1 n i n i + 1 + n i ) 2 δ ( z l ( x , y ) ) d z = NA 2 2 0 L R S ( x , y , z ) d z .

Metrics