Abstract

Two-photon laser scanning microscopy (2PLSM) is an important tool for in vivo tissue imaging with sub-cellular resolution, but the penetration depth of current systems is potentially limited by sample-induced optical aberrations. To quantify these, we measured the refractive index n' in the somatosensory cortex of 7 rats in vivo using defocus optimization in full-field optical coherence tomography (ff-OCT). We found n' to be independent of imaging depth or rat age. From these measurements, we calculated that two-photon imaging beyond 200µm into the cortex is limited by spherical aberration, indicating that adaptive optics will improve imaging depth.

©2011 Optical Society of America

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2009 (4)

2008 (1)

2007 (4)

2006 (2)

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[Crossref] [PubMed]

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600 nm,” Phys. Med. Biol. 51(6), 1479–1489 (2006).
[Crossref] [PubMed]

2005 (1)

2003 (3)

2002 (2)

2001 (1)

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[Crossref] [PubMed]

2000 (1)

A. Knuettel and M. Boehlau-Godau, “Spatially confined and temporally resolved refractive index and scattering evaluation in human skin performed with optical coherence tomography,” J. Biomed. Opt. 5(1), 83–92 (2000).
[Crossref]

1999 (1)

A. Egner and S. W. Hell, “Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch,” J. Microsc. 193(3), 244–249 (1999).
[Crossref]

1998 (1)

S. N. Roper, M. D. Moores, G. V. Gelikonov, F. I. Feldchtein, N. M. Beach, M. A. King, V. M. Gelikonov, A. M. Sergeev, and D. H. Reitze, “In vivo detection of experimentally induced cortical dysgenesis in the adult rat neocortex using optical coherence tomography,” J. Neurosci. Methods 80(1), 91–98 (1998).
[Crossref] [PubMed]

1997 (1)

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, “In vivo dendritic calcium dynamics in neocortical pyramidal neurons,” Nature 385(6612), 161–165 (1997).
[Crossref] [PubMed]

1996 (2)

1995 (3)

Alexandrov, S.

Alexandrov, S. A.

Armstrong, J.

Badizadegan, K.

Beach, N. M.

S. N. Roper, M. D. Moores, G. V. Gelikonov, F. I. Feldchtein, N. M. Beach, M. A. King, V. M. Gelikonov, A. M. Sergeev, and D. H. Reitze, “In vivo detection of experimentally induced cortical dysgenesis in the adult rat neocortex using optical coherence tomography,” J. Neurosci. Methods 80(1), 91–98 (1998).
[Crossref] [PubMed]

Beaurepaire, E.

A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, “High-resolution full-field optical coherence tomography with a Linnik microscope,” Appl. Opt. 41(4), 805–812 (2002).
[Crossref] [PubMed]

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[Crossref] [PubMed]

Betzig, E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2009).
[Crossref] [PubMed]

Bewersdorf, J.

Boccara, A. C.

Boehlau-Godau, M.

A. Knuettel and M. Boehlau-Godau, “Spatially confined and temporally resolved refractive index and scattering evaluation in human skin performed with optical coherence tomography,” J. Biomed. Opt. 5(1), 83–92 (2000).
[Crossref]

Booker, G. R.

Booth, M. J.

Boppart, S. A.

G. J. Tearney, B. E. Bouma, S. A. Boppart, B. Golubovic, E. A. Swanson, and J. G. Fujimoto, “Rapid acquisition of in vivo biological images by use of optical coherence tomography,” Opt. Lett. 21(17), 1408–1410 (1996).
[Crossref] [PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. Bouma, M. R. Hee, J. F. Southern, and E. A. Swanson, “Optical biopsy and imaging using optical coherence tomography,” Nat. Med. 1(9), 970–972 (1995).
[Crossref] [PubMed]

Botcherby, E. J.

Bouma, B.

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. Bouma, M. R. Hee, J. F. Southern, and E. A. Swanson, “Optical biopsy and imaging using optical coherence tomography,” Nat. Med. 1(9), 970–972 (1995).
[Crossref] [PubMed]

Bouma, B. E.

Brezinski, M. E.

G. J. Tearney, M. E. Brezinski, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto, “Determination of the refractive index of highly scattering human tissue by optical coherence tomography,” Opt. Lett. 20(21), 2258–2260 (1995).
[Crossref] [PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. Bouma, M. R. Hee, J. F. Southern, and E. A. Swanson, “Optical biopsy and imaging using optical coherence tomography,” Nat. Med. 1(9), 970–972 (1995).
[Crossref] [PubMed]

Chaigneau, E.

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[Crossref] [PubMed]

Charpak, S.

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[Crossref] [PubMed]

Cuche, E.

Daimon, M.

Dasari, R. R.

David, G.

Debarre, D.

Débarre, D.

Denk, W.

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[Crossref] [PubMed]

P. Theer, M. T. Hasan, and W. Denk, “Two-photon imaging to a depth of 1000 microm in living brains by use of a Ti:Al2O3 regenerative amplifier,” Opt. Lett. 28(12), 1022–1024 (2003).
[Crossref] [PubMed]

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, “In vivo dendritic calcium dynamics in neocortical pyramidal neurons,” Nature 385(6612), 161–165 (1997).
[Crossref] [PubMed]

Depeursinge, C.

Ding, H.

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600 nm,” Phys. Med. Biol. 51(6), 1479–1489 (2006).
[Crossref] [PubMed]

Dubois, A.

Durst, M. E.

Egner, A.

A. Egner and S. W. Hell, “Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch,” J. Microsc. 193(3), 244–249 (1999).
[Crossref]

Emery, Y.

Feld, M. S.

Feldchtein, F. I.

S. N. Roper, M. D. Moores, G. V. Gelikonov, F. I. Feldchtein, N. M. Beach, M. A. King, V. M. Gelikonov, A. M. Sergeev, and D. H. Reitze, “In vivo detection of experimentally induced cortical dysgenesis in the adult rat neocortex using optical coherence tomography,” J. Neurosci. Methods 80(1), 91–98 (1998).
[Crossref] [PubMed]

Fercher, A. F.

A. F. Fercher, “Optical Coherence Tomography,” J. Biomed. Opt. 1(2), 157–173 (1996).
[Crossref]

Fujimoto, J. G.

Gelikonov, G. V.

S. N. Roper, M. D. Moores, G. V. Gelikonov, F. I. Feldchtein, N. M. Beach, M. A. King, V. M. Gelikonov, A. M. Sergeev, and D. H. Reitze, “In vivo detection of experimentally induced cortical dysgenesis in the adult rat neocortex using optical coherence tomography,” J. Neurosci. Methods 80(1), 91–98 (1998).
[Crossref] [PubMed]

Gelikonov, V. M.

S. N. Roper, M. D. Moores, G. V. Gelikonov, F. I. Feldchtein, N. M. Beach, M. A. King, V. M. Gelikonov, A. M. Sergeev, and D. H. Reitze, “In vivo detection of experimentally induced cortical dysgenesis in the adult rat neocortex using optical coherence tomography,” J. Neurosci. Methods 80(1), 91–98 (1998).
[Crossref] [PubMed]

Georges, P.

Gigan, S.

Golubovic, B.

Hasan, M. T.

Hee, M. R.

G. J. Tearney, M. E. Brezinski, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto, “Determination of the refractive index of highly scattering human tissue by optical coherence tomography,” Opt. Lett. 20(21), 2258–2260 (1995).
[Crossref] [PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. Bouma, M. R. Hee, J. F. Southern, and E. A. Swanson, “Optical biopsy and imaging using optical coherence tomography,” Nat. Med. 1(9), 970–972 (1995).
[Crossref] [PubMed]

Hell, S. W.

A. Egner and S. W. Hell, “Equivalence of the Huygens-Fresnel and Debye approach for the calculation of high aperture point-spread functions in the presence of refractive index mismatch,” J. Microsc. 193(3), 244–249 (1999).
[Crossref]

Hillman, T.

Hu, X. H.

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600 nm,” Phys. Med. Biol. 51(6), 1479–1489 (2006).
[Crossref] [PubMed]

Ji, N.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2009).
[Crossref] [PubMed]

Juskaitis, R.

King, M. A.

S. N. Roper, M. D. Moores, G. V. Gelikonov, F. I. Feldchtein, N. M. Beach, M. A. King, V. M. Gelikonov, A. M. Sergeev, and D. H. Reitze, “In vivo detection of experimentally induced cortical dysgenesis in the adult rat neocortex using optical coherence tomography,” J. Neurosci. Methods 80(1), 91–98 (1998).
[Crossref] [PubMed]

Kleinfeld, D.

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, “In vivo dendritic calcium dynamics in neocortical pyramidal neurons,” Nature 385(6612), 161–165 (1997).
[Crossref] [PubMed]

Knuettel, A.

A. Knuettel and M. Boehlau-Godau, “Spatially confined and temporally resolved refractive index and scattering evaluation in human skin performed with optical coherence tomography,” J. Biomed. Opt. 5(1), 83–92 (2000).
[Crossref]

Kobat, D.

Kragel, P. J.

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600 nm,” Phys. Med. Biol. 51(6), 1479–1489 (2006).
[Crossref] [PubMed]

Labiau, S.

Laczik, Z.

Lessard, M. D.

Lu, J. Q.

H. Ding, J. Q. Lu, W. A. Wooden, P. J. Kragel, and X. H. Hu, “Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600 nm,” Phys. Med. Biol. 51(6), 1479–1489 (2006).
[Crossref] [PubMed]

Lue, N.

Mack-Bucher, J. A.

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[Crossref] [PubMed]

Magistretti, P.

Marquet, P.

Masumura, A.

Mertz, J.

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[Crossref] [PubMed]

Milkie, D. E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7(2), 141–147 (2009).
[Crossref] [PubMed]

Moores, M. D.

S. N. Roper, M. D. Moores, G. V. Gelikonov, F. I. Feldchtein, N. M. Beach, M. A. King, V. M. Gelikonov, A. M. Sergeev, and D. H. Reitze, “In vivo detection of experimentally induced cortical dysgenesis in the adult rat neocortex using optical coherence tomography,” J. Neurosci. Methods 80(1), 91–98 (1998).
[Crossref] [PubMed]

Moreau, J.

Nishimura, N.

Oheim, M.

M. Oheim, E. Beaurepaire, E. Chaigneau, J. Mertz, and S. Charpak, “Two-photon microscopy in brain tissue: parameters influencing the imaging depth,” J. Neurosci. Methods 111(1), 29–37 (2001).
[Crossref] [PubMed]

Popescu, G.

Rappaz, B.

Reitze, D. H.

S. N. Roper, M. D. Moores, G. V. Gelikonov, F. I. Feldchtein, N. M. Beach, M. A. King, V. M. Gelikonov, A. M. Sergeev, and D. H. Reitze, “In vivo detection of experimentally induced cortical dysgenesis in the adult rat neocortex using optical coherence tomography,” J. Neurosci. Methods 80(1), 91–98 (1998).
[Crossref] [PubMed]

Roper, S. N.

S. N. Roper, M. D. Moores, G. V. Gelikonov, F. I. Feldchtein, N. M. Beach, M. A. King, V. M. Gelikonov, A. M. Sergeev, and D. H. Reitze, “In vivo detection of experimentally induced cortical dysgenesis in the adult rat neocortex using optical coherence tomography,” J. Neurosci. Methods 80(1), 91–98 (1998).
[Crossref] [PubMed]

Rueckel, M.

M. Rueckel, J. A. Mack-Bucher, and W. Denk, “Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,” Proc. Natl. Acad. Sci. U.S.A. 103(46), 17137–17142 (2006).
[Crossref] [PubMed]

Sacchet, D.

Sampson, D.

Sampson, D. D.

Schaffer, C. B.

Sergeev, A. M.

S. N. Roper, M. D. Moores, G. V. Gelikonov, F. I. Feldchtein, N. M. Beach, M. A. King, V. M. Gelikonov, A. M. Sergeev, and D. H. Reitze, “In vivo detection of experimentally induced cortical dysgenesis in the adult rat neocortex using optical coherence tomography,” J. Neurosci. Methods 80(1), 91–98 (1998).
[Crossref] [PubMed]

Silva, K. K.

Silva, K. K. M. B.

Southern, J. F.

G. J. Tearney, M. E. Brezinski, J. F. Southern, B. E. Bouma, M. R. Hee, and J. G. Fujimoto, “Determination of the refractive index of highly scattering human tissue by optical coherence tomography,” Opt. Lett. 20(21), 2258–2260 (1995).
[Crossref] [PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. Bouma, M. R. Hee, J. F. Southern, and E. A. Swanson, “Optical biopsy and imaging using optical coherence tomography,” Nat. Med. 1(9), 970–972 (1995).
[Crossref] [PubMed]

Srinivas, S.

Svoboda, K.

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, “In vivo dendritic calcium dynamics in neocortical pyramidal neurons,” Nature 385(6612), 161–165 (1997).
[Crossref] [PubMed]

Swanson, E. A.

G. J. Tearney, B. E. Bouma, S. A. Boppart, B. Golubovic, E. A. Swanson, and J. G. Fujimoto, “Rapid acquisition of in vivo biological images by use of optical coherence tomography,” Opt. Lett. 21(17), 1408–1410 (1996).
[Crossref] [PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. Bouma, M. R. Hee, J. F. Southern, and E. A. Swanson, “Optical biopsy and imaging using optical coherence tomography,” Nat. Med. 1(9), 970–972 (1995).
[Crossref] [PubMed]

Tank, D. W.

K. Svoboda, W. Denk, D. Kleinfeld, and D. W. Tank, “In vivo dendritic calcium dynamics in neocortical pyramidal neurons,” Nature 385(6612), 161–165 (1997).
[Crossref] [PubMed]

Tearney, G. J.

Theer, P.

Török, P.

Tsuzuki, T.

Vabre, L.

Varga, P.

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Supplementary Material (2)

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Figures (6)

Fig. 1
Fig. 1 Ff- OCT setup and sample preparation. Light from a low-coherence source enters a Linnik interferometer. The sample arm consists of a microscope objective on a motorized linear translation stage allowing axial movement and of the sample, which is equally translatable along the optical axis. The animal is held by a metal head fixation plate glued to its skull around the craniotomy, which consists of a cover slip glued onto the thinned bone around the actual opening, where the brain tissue comes into direct contact with the cover slip. The reference arm consists of a folding mirror, a cover slip to compensate for dispersion from the cover slip on the rat brain, an objective identical to the sample objective and a reference mirror mounted on a piezo actuator for phase-stepping. The focal planes of both objectives are imaged using a lens doublet serving as tube lens onto an InGaAs camera. The piezo, camera and both motorized stages are controlled by a standard PC running Light-CT software.

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Fig. 2
Fig. 2 (Color online). Defocus caused by refractive index mismatch in high-NA OCT imaging and defocus correction. (a) OCT image of brain surface; the coherence volume coincides with the focus of the objective, leading to good imaging limited only by diffraction and speckle. The relative arm length δ is by definition 0. (b) When imaging at a nominal depth zN of 205µm below the tissue surface, the refractive index mismatch causes the coherence volume to move upwards and the actual focus zA to move downwards with respect to zN. The plane imaged with OCT, defined by the position of the coherence volume (CV), contains hardly any structure due to defocus aberration. This is equally true at relative arm length δ = 0 (not shown) and at the non-optimized relative arm length δ = 10.8µm shown here. (c) Moving the objective by a distance l and simultaneously changing the reference arm length by δ = 2 l (ng −1) allows the actual focus zA to be brought into coincidence with the coherence volume, restoring OCT signal and revealing the spatial reflectivity variations of the sample. Note that this decreases the nominal imaging depth by l to zN= 199µm. (d) OCT signal depends on the correct setting of the reference arm length δ. The data points corresponding to the images shown in a-c are indicated by the corresponding letters. A Gaussian with a baseline is fitted to the data to find the position of the peak, which corresponds to the optimal reference arm length δ at that depth zN. Due to defocus, the optimal reference arm length δ at zN = 199µm has shifted to higher values compared to zN = 0 µm. (e) The spatial frequency content of images for different reference arm lengths δ shows image degradation (loss in low and medium spatial frequency content) when a non-optimal reference arm length introduces defocus.

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Fig. 3
Fig. 3 For three 3 week old (a), two 6 week old (b) and two 12 week old (c) rats, the optimal reference arm length δ (see Fig. 2d) is plotted as a function of nominal imaging depth zN. Each plot corresponds to one lateral position in one rat, e.g. plot “M1-2” corresponds to the second lateral position in the first 6 week old rat. For reference, the theoretical curves corresponding to fixed refractive index values from 1.33 to 1.37 are shown in all plots; the values are given in the upper left hand plot. These reference curves are fairly linear in the range of depths shown. From each plot, the mean slope δ/zN was determined by least squares fitting and the refractive index n' was calculated. All values for different positions in animals of the same age were averaged to give the mean refractive index for animals of that age, shown in the right panel. One-way ANOVA showed no significant correlation between age and refractive index.

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Fig. 4
Fig. 4 Single-frame excerpts from ff-OCT video recordings of surface blood vessels in rat cortex. (Left/Center, Media 1) large vessel of a p46 rat; 6.6 Hz frame rate; produced from a 33 Hz video by averaging 5 ffOCT images each to increase signal to noise. In the upper left hand corner the tissue surrounding the vessel can be seen; passing from lower left to upper center individual leukocytes can be seen moving slowly along the vessel wall (one of them is marked with a white arrow). The right half of the field of view shows the interior of the vessel where objects move too rapidly to be resolved individually. In the lower center of the field of view, the vessel wall descends into the coherence volume so that leukocytes can be seen as if through a semi-transparent wall. (Right, Media 2) junction of two blood vessels joining in the cortex of a p21 rat. 4 seconds at 33Hz frame rate, taken without any averaging. In the upper vessel, an individual object can be seen which moves much slower than the surrounding blood, possibly due to interactions with the blood vessel wall.

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Fig. 5
Fig. 5 The decrease of OCT signal with depth for different reference arm configurations shows the importance of defocus correction for penetration depth. Only with depth-dependent defocus correction is maximum signal at all depths achieved (thick solid line). Without defocus correction, the OCT signal decreases very rapidly (thick dashed curve). A fixed reference arm length optimized for imaging at depths of 200, 300 and 400µm, respectively, yields optimal signal at that depth but a low signal at all other depths (thin curves). The sample was the upper cortical layers of a young rat, imaged in vivo with ff-OCT at 33Hz. Each data point corresponds to the mean signal of four image frames taken at the same depth, corresponding to 120ms acquisition time. Axial scanning was performed with a 10µm step size.

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Fig. 6
Fig. 6 Consequences of refractive index mismatch for 2PLSM. (a) Schematic drawing of the situation being simulated: a water immersion objective is used to focus a laser beam into a sample with refractive index n' which is larger than the refractive index n of the immersion medium. The aberrated two-photon excitation PSF for imaging at different depths z is calculated and analyzed. (b) Two-photon excitation loss caused by depth aberrations: for a single fluorophore (i.e. maximum of PSF), a fluorescent sheet (brightest xy plane of the PSF) and uniformly stained sample (3D integral of PSF). (c) Loss in axial resolution in 2PLSM (FWHM of response to fluorescent sheet, determined from a Gaussian fit to the simulated PSF). (d) Loss in lateral resolution in 2PLSM (FWHM of response to single fluorophore, determined from a Gaussian fit to the simulated PSF)

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Equations (7)

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I = | E S M P + E R E F | 2 = | E S M P | 2 + | E R E F | 2 + 2 ℜ ( E S M P ∗ E R E F )
E S M P ∗ E R E F = ∫ d λ ∫ 2 π α i n d α i n ∫ 2 π α o u t d α o u t K ( λ , α i n , α o u t )
K ( λ , α i n , α o u t ) = E S M P ∗ E R E F = e − i 2 π λ [ δ 0 + Ψ ( λ , α i n , α o u t ) ] e + i 2 π λ [ δ 0 + δ ] = e − i 2 π λ [ Ψ ( λ , α i n , α o u t ) − δ ]
Ψ ( λ , α i n , α o u t ) = Ψ ( λ , α i n ) + Ψ ( λ , α o u t )
Ψ ( λ , α ) = z N n ( λ ) cos ( α ) − z A n ' ( λ ) cos ( α ' ) = z N n ( λ ) cos ( α ) − z A n ' ( λ ) 2 − n ( λ ) 2 sin 2 ( α )
Ψ ( λ , α i n , α o u t ) = Ψ ( λ , α i n ) + Ψ ( λ , α o u t ) = z N n ( λ ) ( cos ( α i n ) − cos ( α o u t ) ) − z A ( n ' ( λ ) 2 − n ( λ ) 2 sin 2 ( α i n ) − n ' ( λ ) 2 − n ( λ ) 2 sin 2 ( α o u t ) )
n ' m arg i n a l sin { arctan [ n ' m arg i n a l z N tan ( arcsin N A n ) n z N + δ / 2 ] } = N A

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