Abstract

The behavior of two-photon fluorescence imaging through a scattering medium is analyzed by use of the Monte Carlo technique. The axial and transverse distributions of the excitation photons in the focused Gaussian beam are derived for both isotropic and anisotropic scatterers at different numerical apertures and at various ratios of the scattering depth with the mean free path. The two-photon fluorescence profiles of the sample are determined from the square of the normalized excitation intensity distributions. For the same lens aperture and scattering medium, two-photon fluorescence imaging offers a sharper and less aberrated axial response than that of single-photon confocal fluorescence imaging. The contrast in the corresponding transverse fluorescence profile is also significantly higher. Also presented are results comparing the effects of isotropic and anisotropic scattering media in confocal reflection imaging. The convergence properties of the Monte Carlo simulation are also discussed.

© 1998 Optical Society of America

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References

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1998

C. Saloma, C. Palmes-Saloma, H. Kondoh, “Site-specific confocal fluorescence imaging of biological microstructures in a turbid medium,” Phys. Med. Biol. 43, 1741–1759 (1998).
[CrossRef] [PubMed]

1997

W. Denk, K. Svoboda, “Photon upmanship: why multiphoton is more than a gimmick,” Neuron 18, 351–357 (1997).
[CrossRef] [PubMed]

M. Schrader, K. Bahlmann, S. Hell, “Three-photon excitation microscopy: theory, experiment and applications,” Optik 104, 116–121 (1997).

W. Fisher, E. Wachter, M. Armas, C. Seaton, “Ti:sapphire laser as an excitation source in two-photon spectroscopy,” Appl. Spectrosc. 51, 218–226 (1997).
[CrossRef]

1996

1995

1994

1993

O. Nakamura, “Three-dimensional imaging characteristics of laser scan fluorescence microscopy: two-photon excitation versus single-photon excitation,” Optik 93, 39–42 (1993).

B. Rosen, R. Beddington, “Whole-mount hybridization in the mouse embryo: gene expression in three dimensions,” Trends Genet. 9, 162–163 (1993).
[CrossRef] [PubMed]

1990

W. Denk, J. Strickler, W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

C. Sheppard, K. Gu, “Image formation in two-photon fluorescence microscopy,” Optik 86, 104–106 (1990).

W. Cheong, S. Prahl, A. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1989

H. Haugen, A. Orthonos, “Fluorescence studies of multiphoton ionization processes: four- and five-photon ionization of Sr at wavelengths of 558–590 nm,” Phys. Rev. A 39, 3392–3399 (1989).
[CrossRef] [PubMed]

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

1987

S. Flock, B. Wilson, M. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

S. Jacques, C. Alter, S. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Laser Life Sci. 1, 309–333 (1987).

Alter, C.

S. Jacques, C. Alter, S. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Laser Life Sci. 1, 309–333 (1987).

Armas, M.

Bahlmann, K.

M. Schrader, K. Bahlmann, S. Hell, “Three-photon excitation microscopy: theory, experiment and applications,” Optik 104, 116–121 (1997).

Beddington, R.

B. Rosen, R. Beddington, “Whole-mount hybridization in the mouse embryo: gene expression in three dimensions,” Trends Genet. 9, 162–163 (1993).
[CrossRef] [PubMed]

Ben-Letaief, K.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1991).

Cambaliza, M.

Cheng, H.

Cheong, W.

W. Cheong, S. Prahl, A. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Delaney, K.

W. Denk, K. Delaney, A. Gelperin, D. Kleinfeld, B. Strowbridge, D. Tank, R. Yuste, “Anatomical and functional imaging of neurons using two-photon laser scanning microscopy,” J. Neurosci. Methods 54, 151–162 (1994).
[CrossRef] [PubMed]

Denk, W.

W. Denk, K. Svoboda, “Photon upmanship: why multiphoton is more than a gimmick,” Neuron 18, 351–357 (1997).
[CrossRef] [PubMed]

K. Svoboda, W. Denk, S. Tsuda, “Two-photon excitation scanning microscopy of living neurons with a saturable Bragg reflector mode-locked diode pumped Cr:LiSrAlFl laser,” Opt. Lett. 21, 1411–1413 (1996).
[CrossRef] [PubMed]

R. Yuste, W. Denk, “Dendritic spines as basic functional units of neuronal integration,” Nature 375, 682–684 (1995).
[CrossRef] [PubMed]

W. Denk, K. Delaney, A. Gelperin, D. Kleinfeld, B. Strowbridge, D. Tank, R. Yuste, “Anatomical and functional imaging of neurons using two-photon laser scanning microscopy,” J. Neurosci. Methods 54, 151–162 (1994).
[CrossRef] [PubMed]

W. Denk, J. Strickler, W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Dunn, A.

Fisher, W.

Flock, S.

S. Flock, B. Wilson, M. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Gelperin, A.

W. Denk, K. Delaney, A. Gelperin, D. Kleinfeld, B. Strowbridge, D. Tank, R. Yuste, “Anatomical and functional imaging of neurons using two-photon laser scanning microscopy,” J. Neurosci. Methods 54, 151–162 (1994).
[CrossRef] [PubMed]

Gu, K.

C. Sheppard, K. Gu, “Image formation in two-photon fluorescence microscopy,” Optik 86, 104–106 (1990).

Gu, M.

Haugen, H.

H. Haugen, A. Orthonos, “Fluorescence studies of multiphoton ionization processes: four- and five-photon ionization of Sr at wavelengths of 558–590 nm,” Phys. Rev. A 39, 3392–3399 (1989).
[CrossRef] [PubMed]

Haugland, R.

R. Haugland, Handbook in Fluorescent Probes and Research Chemicals, 6th ed. (Molecular Probes, Eugene, Ore., 1996).

Hell, S.

M. Schrader, K. Bahlmann, S. Hell, “Three-photon excitation microscopy: theory, experiment and applications,” Optik 104, 116–121 (1997).

Jacques, S.

S. Jacques, C. Alter, S. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Laser Life Sci. 1, 309–333 (1987).

Kirby, M. S.

Kleinfeld, D.

W. Denk, K. Delaney, A. Gelperin, D. Kleinfeld, B. Strowbridge, D. Tank, R. Yuste, “Anatomical and functional imaging of neurons using two-photon laser scanning microscopy,” J. Neurosci. Methods 54, 151–162 (1994).
[CrossRef] [PubMed]

Knuttel, A.

Kondoh, H.

C. Saloma, C. Palmes-Saloma, H. Kondoh, “Site-specific confocal fluorescence imaging of biological microstructures in a turbid medium,” Phys. Med. Biol. 43, 1741–1759 (1998).
[CrossRef] [PubMed]

Lederer, W.

Manni, J.

J. Manni, “Two-photon excitation expands the capabilities of laser-scanning microscopy,” Biophotonics 1996, 44–49.

Manolakis, D.

J. Proakis, D. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1989), pp. 129–133.

Nakamura, O.

O. Nakamura, “Three-dimensional imaging characteristics of laser scan fluorescence microscopy: two-photon excitation versus single-photon excitation,” Optik 93, 39–42 (1993).

Newton, R.

R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

Orthonos, A.

H. Haugen, A. Orthonos, “Fluorescence studies of multiphoton ionization processes: four- and five-photon ionization of Sr at wavelengths of 558–590 nm,” Phys. Rev. A 39, 3392–3399 (1989).
[CrossRef] [PubMed]

Palmes-Saloma, C.

C. Saloma, C. Palmes-Saloma, H. Kondoh, “Site-specific confocal fluorescence imaging of biological microstructures in a turbid medium,” Phys. Med. Biol. 43, 1741–1759 (1998).
[CrossRef] [PubMed]

Patterson, M.

S. Flock, B. Wilson, M. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Patterson, M. S.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Piston, D.

Potter, S.

S. Potter, “Vital imaging: two photons are better than one,” Current Biol. 6, 1595–1598 (1996).
[CrossRef]

Prahl, S.

W. Cheong, S. Prahl, A. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

S. Jacques, C. Alter, S. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Laser Life Sci. 1, 309–333 (1987).

Proakis, J.

J. Proakis, D. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1989), pp. 129–133.

Richards-Kortum, R.

Rosen, B.

B. Rosen, R. Beddington, “Whole-mount hybridization in the mouse embryo: gene expression in three dimensions,” Trends Genet. 9, 162–163 (1993).
[CrossRef] [PubMed]

Saloma, C.

C. Saloma, C. Palmes-Saloma, H. Kondoh, “Site-specific confocal fluorescence imaging of biological microstructures in a turbid medium,” Phys. Med. Biol. 43, 1741–1759 (1998).
[CrossRef] [PubMed]

M. Cambaliza, C. Saloma, “Single Gaussian beam interaction with a dielectric microsphere: radiation forces, multiple internal reflections, and caustic structures,” Appl. Opt. 34, 3522–3528 (1995).
[CrossRef]

Schmitt, J.

Schrader, M.

M. Schrader, K. Bahlmann, S. Hell, “Three-photon excitation microscopy: theory, experiment and applications,” Optik 104, 116–121 (1997).

Seaton, C.

Seigman, A.

A. Seigman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.

Sheppard, C.

Smithpeter, C.

Strickler, J.

W. Denk, J. Strickler, W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Strowbridge, B.

W. Denk, K. Delaney, A. Gelperin, D. Kleinfeld, B. Strowbridge, D. Tank, R. Yuste, “Anatomical and functional imaging of neurons using two-photon laser scanning microscopy,” J. Neurosci. Methods 54, 151–162 (1994).
[CrossRef] [PubMed]

Svoboda, K.

Tank, D.

W. Denk, K. Delaney, A. Gelperin, D. Kleinfeld, B. Strowbridge, D. Tank, R. Yuste, “Anatomical and functional imaging of neurons using two-photon laser scanning microscopy,” J. Neurosci. Methods 54, 151–162 (1994).
[CrossRef] [PubMed]

Tannous, T.

Tsuda, S.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Scattering Light Tables, Formulas and Applications (Academic, New York, 1980).

Wachter, E.

Webb, W.

Welch, A.

A. Dunn, C. Smithpeter, A. Welch, R. Richards-Kortum, “Sources of contrast in confocal reflectance imaging,” Appl. Opt. 35, 3441–3446 (1996).
[CrossRef] [PubMed]

W. Cheong, S. Prahl, A. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Wilkinson, D.

D. Wilkinson, In Situ Hybridization: A Practical Approach (International Reproduction Ltd., New York, 1993), pp. 75–83.

Wilson, B.

S. Flock, B. Wilson, M. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Wilson, B. C.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1991).

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Xu, C.

Yadlowsky, M.

Yuste, R.

R. Yuste, W. Denk, “Dendritic spines as basic functional units of neuronal integration,” Nature 375, 682–684 (1995).
[CrossRef] [PubMed]

W. Denk, K. Delaney, A. Gelperin, D. Kleinfeld, B. Strowbridge, D. Tank, R. Yuste, “Anatomical and functional imaging of neurons using two-photon laser scanning microscopy,” J. Neurosci. Methods 54, 151–162 (1994).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Spectrosc.

Biophotonics

J. Manni, “Two-photon excitation expands the capabilities of laser-scanning microscopy,” Biophotonics 1996, 44–49.

Current Biol.

S. Potter, “Vital imaging: two photons are better than one,” Current Biol. 6, 1595–1598 (1996).
[CrossRef]

IEEE J. Quantum Electron.

W. Cheong, S. Prahl, A. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

IEEE Trans. Biomed. Eng.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in highly scattering tissues—I,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

J. Neurosci. Methods

W. Denk, K. Delaney, A. Gelperin, D. Kleinfeld, B. Strowbridge, D. Tank, R. Yuste, “Anatomical and functional imaging of neurons using two-photon laser scanning microscopy,” J. Neurosci. Methods 54, 151–162 (1994).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Laser Life Sci.

S. Jacques, C. Alter, S. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Laser Life Sci. 1, 309–333 (1987).

Med. Phys.

S. Flock, B. Wilson, M. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Nature

R. Yuste, W. Denk, “Dendritic spines as basic functional units of neuronal integration,” Nature 375, 682–684 (1995).
[CrossRef] [PubMed]

Neuron

W. Denk, K. Svoboda, “Photon upmanship: why multiphoton is more than a gimmick,” Neuron 18, 351–357 (1997).
[CrossRef] [PubMed]

Opt. Lett.

Optik

M. Schrader, K. Bahlmann, S. Hell, “Three-photon excitation microscopy: theory, experiment and applications,” Optik 104, 116–121 (1997).

C. Sheppard, K. Gu, “Image formation in two-photon fluorescence microscopy,” Optik 86, 104–106 (1990).

O. Nakamura, “Three-dimensional imaging characteristics of laser scan fluorescence microscopy: two-photon excitation versus single-photon excitation,” Optik 93, 39–42 (1993).

Phys. Med. Biol.

C. Saloma, C. Palmes-Saloma, H. Kondoh, “Site-specific confocal fluorescence imaging of biological microstructures in a turbid medium,” Phys. Med. Biol. 43, 1741–1759 (1998).
[CrossRef] [PubMed]

Phys. Rev. A

H. Haugen, A. Orthonos, “Fluorescence studies of multiphoton ionization processes: four- and five-photon ionization of Sr at wavelengths of 558–590 nm,” Phys. Rev. A 39, 3392–3399 (1989).
[CrossRef] [PubMed]

Science

W. Denk, J. Strickler, W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Trends Genet.

B. Rosen, R. Beddington, “Whole-mount hybridization in the mouse embryo: gene expression in three dimensions,” Trends Genet. 9, 162–163 (1993).
[CrossRef] [PubMed]

Other

D. Wilkinson, In Situ Hybridization: A Practical Approach (International Reproduction Ltd., New York, 1993), pp. 75–83.

R. Haugland, Handbook in Fluorescent Probes and Research Chemicals, 6th ed. (Molecular Probes, Eugene, Ore., 1996).

R. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966).

A. Seigman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.

J. Proakis, D. Manolakis, Introduction to Digital Signal Processing (Macmillan, New York, 1989), pp. 129–133.

H. C. van de Hulst, Multiple Scattering Light Tables, Formulas and Applications (Academic, New York, 1980).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1991).

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Figures (15)

Fig. 1
Fig. 1

Optical configuration of a confocal fluorescence microscope used in MC simulation. Lenses L1, L2, and L3 are identical. The direction of the scattered photon in the scattering medium of refractive index n is given by angles θ and φ.

Fig. 2
Fig. 2

Plot of E versus h/ d s , where E = N E /N e , N E is the number of incident photons reaching the focal spot area πw 0 2 in the sample plane, N e = 10, n = 1.57, g = 0 (isotropic), g = 0.9 (anisotropic). Different NA values of 0.5, 0.7, and 1 are considered for L1.

Fig. 3
Fig. 3

Confocal reflection imaging. Dependence of R av with pinhole radius p for h/ d s = 1 (squares), 2 (solid squares), 3 (circles), and 4 (cross hairs): (a) g = 0.9, (b) g - 0, (c) g = 0 (for small p values). Parameters: strip width is 65 μm, n = 1, N e = 107, w 0 = 0.15 μm, and NA ≈ 0.7.

Fig. 4
Fig. 4

Confocal imaging of perfectly reflecting strips: (a) h/ d s = 0, (b) g = 0 (h/ d s = 4), (c) g = 0.9 (h/ d s = 4). Parameters: p = w 0 = 2.5 μm, NA ≈ 0.7, N e = 107, sampling interval is 1 μm, and n = 1. Choice of p = 2.5 μm yields the best results in terms of signal-to-noise ratio and minimum loss of imaging resolution. It also allows for a reasonable length of computation time. (One data set takes 6 h to generate on an Alpha computer running at 533 MHz).

Fig. 5
Fig. 5

Flow chart of MC simulation for describing the emission of a fluorescence photon by an excitation photon that has reached the sample plane through the scattering medium.

Fig. 6
Fig. 6

Flow chart of MC simulation for describing the possible detection of a fluorescence photon that propagates from the sample plane through the turbid medium. Each fluorescence photon is released at a uniformly random direction within the (upper) hemispherical solid angle centered about the emitting point in the sample plane.

Fig. 7
Fig. 7

Confocal fluorescence imaging. Fluorescence photon fraction F av versus pinhole radius p for h/ d s = 1 (squares), 2 (solid squares), 3 (circles), and 4 mm (cross hairs): (a) g = 0.9 and (b) g = 0. Parameters: n = 1, F = N f det/N e , N e = 107, w 0 = 0.15 μm, NA ≈ 0.7, and number of trials is 10.

Fig. 8
Fig. 8

Confocal fluorescence imaging. Behavior of signal-to-noise ratio ρ with pinhole radius p for h/ d s = 1 (cross hairs), 2 (solid squares), 3 (circles), and 4 mm (solid circles): (a) g = 0.9 and (b) g = 0, where ρ = F av(p)/lsdl, sd is the standard deviation (F = N f det/N e ), N e = 107, w 0 = 0.15 μm, and NA ≈ 0.7.

Fig. 9
Fig. 9

Confocal fluorescence imaging. Normalized axial-intensity profiles for various h/ d s values (p = 0.25 mm, n =1, w 0= 0.15 μm, NA ≈ 0.7, n = 1, N e = 107).

Fig. 10
Fig. 10

Confocal fluorescence imaging. Normalized axial-intensity distributions for various p values (h/ d s = 1, n = 1, w 0 = 0.15 μm, NA ≈ 0.7, n = 1, N e = 107). Also shown in (a) is the axial response (circles) for h/ d s = 0 and p = 0.01 mm.

Fig. 11
Fig. 11

2P excitation. Normalized transverse E(x) distributions at z = 0 for different (h/ d s 2P) values: (a) g = 0.9 and (b) g = 0, where n = 1, w 0 = 0.3 μm, NA ≈ 0.7, N e = 107, sampling interval is 0.016 mm, and z = 0.

Fig. 12
Fig. 12

2P fluorescence F(x) profiles for different (h/ d s 2P) values: (a) g = 0.9 and (b) g = 0, where sampling distance is 0.008 μm, n = 1, w 0 2P = 0.3 μm, NA ≈ 0.7, and N e = 107. The fluorescence profile is given by the square of the corresponding E(x) distribution in Fig. 11. Also shown in (a) is the 1PEF profile for h/ d s 1P = 4.

Fig. 13
Fig. 13

Axial-intensity distributions of the 2P excitation beam (λ E 2P = 700 nm) for various d s 2P values: (a) g = 0.9 and (b) g = 0 (w 0= 0.3 μm, n = 1, NA ≈ 0.7, and N e = 107).

Fig. 14
Fig. 14

2PEF axial-intensity distributions (λ E 2P = 700 nm) for various d s 2P values: (a) g = 0.9 and (b) g = 0 (NA ≈ 0.7, w 0 = 0.3 μm, and N e = 107). The fluorescence profile is given by the square of the corresponding distribution of the axial excitation intensity in Fig. 13.

Fig. 15
Fig. 15

Stability of simulation results (confocal reflection imaging): (a) ρ versus N e for h/ d s = 2 [g = 0.9 (squares), g = 0 (circles)] and 4 [g = 0.9 (solid squares), g = 0 (solid circles)]; (b) ρ versus (h/ d s ) for N e = 107, g = 0 (solid circles), and g = 0.9 (cross hairs). Parameters: number of trials is 10, w 0 = 0.15 μm, p = 2.5 μm, NA = 0.7, and n = 1.

Equations (6)

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d = - d s   ln   Ξ ,
cos   θ = 1 + g 2 2 g - 1 - 2 g - 1 1 - g 2 2 1 - g + 2 g Ξ - 2 ,
0 θ 0 θ A = sin - 1 r 1 / 2 n f 2 + r 1 2 1 / 2 ,
x 1 = A   cos   α ,
y 1 = A   sin   α ,
E NA   =   0.5   =   0.993   exp - 2.331 h / d s ,   E NA   =   0.7   =   1.021   exp - 2.382 h / d s ,   E NA   =   1   =   0.989   exp - 2.426 h / d s .

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