Abstract

Molecular contrast optical coherence tomography (MCOCT) is an extension of OCT in which contrast resulting from the interaction of light with a contrast agent, leads to the enhanced visualization of a specific morphology or biochemical process in a target specimen. In order to improve the sensitivity and specificity of MCOCT, several spectroscopic techniques have recently been introduced which depend upon coherent detection of scattered light which has been modified by interaction with the molecules of interest in a sample. These techniques include harmonic generation, coherent anti-Stokes Raman scattering, linear absorption, and several different forms of pump-probe spectroscopy. We have developed a theoretical framework to facilitate the comparison of the sensitivity of different MCOCT techniques. This framework is based upon the observation that since the noise floor is defined by the reference field power in a shot-noise limited OCT system, the relevant comparison among the techniques is isolated to the available molecular contrast signal power and the algorithm used to extract the signal. We have derived theoretical expressions for the signal power and signal-to-noise ratio for the MCOCT techniques described in the literature based on molecular spectroscopy, as well as several new techniques introduced here.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]

2005 (5)

2004 (9)

N. Nassif, B. Cense, B. H. Park, S. H. Yun, T. C. Chen, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography,” Opt. Lett. 29, 480–482 (2004).
[CrossRef] [PubMed]

B. Hermann, K. K. Bizheva, A. Unterhuber, B. Považay, H. Sattmann, L. Schmetterer, A. F. Fercher, and W. Drexler, “Precision of extracting absorption profiles from weakly scattering media with spectroscopic time-domain optical coherence tomography,” Opt. Express 12, 1677–1688 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1677.
[CrossRef] [PubMed]

Y. Jiang, I. Tomov, Y. Wang, and Z. Chen, “Second-harmonic optical coherence tomography,” Opt. Lett. 29, 1090–1092 (2004).
[CrossRef] [PubMed]

C. Yang, M. A. Choma, L. E. Lamb, J. D. Simon, and J. A. Izatt, “Protein-based molecular contrast optical coherence tomography with phytochrome as the contrast agent,” Opt. Lett. 29, 1396–1398 (2004).
[CrossRef] [PubMed]

C. Yang, L. E. L. McGuckin, J. D. Simon, M. A. Choma, B. E. Applegate, and J. A. Izatt, “Spectral triangulation molecular contrast optical coherence tomography with indocyanine green as the contrast agent,” Opt. Lett. 29, 2016–2018 (2004).
[CrossRef] [PubMed]

B. E. Applegate, C. Yang, A. M. Rollins, and J. A. Izatt, “Polarization resolved second harmonic generation optical coherence tomography in collagen,” Opt. Lett. 29, 2252–2254 (2004).
[CrossRef] [PubMed]

C. Xu, D. L. Marks, M. N. Do, and S. A. Boppart, “Separation of absorption and scattering profiles in spectroscopic optical coherence tomography using a least-squares algorithm,” Opt. Express 12, 4790–4803 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4790.
[CrossRef] [PubMed]

C. Vinegoni, J. S. Bredfeldt, D. L. Marks, and S. A. Boppart, “Nonlinear optical coherence enhancement for optical coherence tomography,” Opt. Express 12, 331–341 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-2-331.
[CrossRef] [PubMed]

D. L. Marks and S. A. Boppart, “Nonlinear Interferometric Vibrational Imaging,” Phys. Rev. Lett. 92, 1239051–1239054 (2004).
[CrossRef]

2003 (8)

P. Stoller, P. M. Celliers, K. M. Reiser, and A. M. Rubenchik, “Quantitative Second-Harmonic Generation Microscopy in Collagen,” Appl. Optics 42, 5209 (2003).
[CrossRef]

W. Mohler, A. C. Millard, and P. J. Campagnola, “Second Harmonic Generation Imaging of Endogenous Structural Proteins,” Methods 29, 97–109 (2003).
[CrossRef] [PubMed]

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300, 1434╍1436 (2003).
[CrossRef] [PubMed]

M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, “Sensitivity advantage of swept-source and Fourier-domain optical coherence tomography,” Opt. Express 11, 2183–2189 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2183.
[CrossRef] [PubMed]

K. D. Rao, M. A. Choma, S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “Molecular contrast in optical coherence tomography by use of a pump-probe technique,” Opt. Lett. 28, 340–342 (2003).
[CrossRef] [PubMed]

D. J. Faber, E. G. Mik, M. C. G. Aalders, and T. G. van Leeuwen, “Light absorption of (oxy-)hemoglobin assessed by spectroscopic optical coherence tomography,” Opt. Lett. 28, 1436–1438 (2003).
[CrossRef] [PubMed]

T. M. Lee, A. L. Oldenburg, S. Sitafalwalla, D. L. Marks, W. Luo, F. J.-J. Toublan, K. S. Suslick, and S. A. Boppart, “Engineered Microsphere Contrast Agents for Optical Coherence Tomography,” Opt. Lett. 28, 1546–1548 (2003).
[CrossRef] [PubMed]

N. Srinivas, V. Rao, and N. Rao, “Saturable and reverse saturable absorption of Rhodamine B in methanol and water,” J. Opt. Soc. Am. B 20, 2470–2479 (2003).
[CrossRef]

2002 (2)

P. F. Tian and W. S. Warren, “Ultrafast measurement of two-photon absorption by loss modulation,” Opt. Lett. 27, 1634╍1636 (2002).
[CrossRef]

T. M. Yelbuz, M. A. Choma, L. Thrane, M. L. Kirby, and J. A. Izatt, “Optical coherence tomography - A new high-resolution imaging technology to study cardiac development in chick embryos,” Circulation 106, 2771–2774 (2002).
[CrossRef] [PubMed]

2001 (2)

S. Radhakrishnan, A. M. Rollins, J. E. Roth, S. Yazdanfar, V. Westphal, D. S. Bardenstein, and J. A. Izatt, “Real-time optical coherence tomography of the anterior segment at 1310 nm,” Arch. Ophthalmol. 119, 1179–1185 (2001).
[PubMed]

A. A. Heikal, S. T. Hess, and W. W. Webb, “Multiphoton molecular spectroscopy and excited-state dynamics of enhanced green fluorescent protein (EGFP): acid-base specificity,” Chem. Phys. 274, 37–55 (2001).
[CrossRef]

1999 (2)

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25, 111–113 (1999).
[CrossRef]

J. K. Barton, J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, and A. J. Welch, “Three-dimensional reconstruction of blood vessels from in vivo color Doppler optical coherence tomography images,” Dermatology 198, 355–361 (1999).
[CrossRef]

1998 (2)

S. Delysse, J.-M. Nunzi, and C. Scala-Valero, “Picosecond optical Kerr ellipsometry determination of S1-Sn absorption spetra of xanthene dyes,” Appl. Phys. B 66, 439–444 (1998).
[CrossRef]

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

1996 (2)

S. A. Boppart, B. E. Bouma, M. E. Brezinski, G. J. Tearney, and J. G. Fujimoto, “Imaging developing neural morphology using optical coherence tomography.,” J. Neurosci. Methods 70, 65–72 (1996), http://research.bmn.com/medline/search/results?uid=MDLN.97137646.
[CrossRef] [PubMed]

S. Reindl and A. Penzkofer, “Triplet quantum yield determination by picosecond laser double-laser fluorescence excitation,” Chem. Phys. 213, 429–438 (1996).
[CrossRef]

1995 (1)

C. Tanielian and C. Wolff, “Determination of the Parameters Controlling Singlet Oxygen Production via Oxygen and Heavy-Atom Enhancement of Triplet Yields,” J. Phys. Chem. 99, 9831–9837 (1995).
[CrossRef]

1977 (1)

A. Owyoung and P. S. Peercy, “Precise Characterixation of the Raman nonlinearity in benzene using nonlinear interferometry,” J. Appl. Physics 48, 674–677 (1977).
[CrossRef]

1976 (1)

M. L. J. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light- absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

Aalders, M. C. G.

Applegate, B. E.

Au, L.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Bardenstein, D. S.

S. Radhakrishnan, A. M. Rollins, J. E. Roth, S. Yazdanfar, V. Westphal, D. S. Bardenstein, and J. A. Izatt, “Real-time optical coherence tomography of the anterior segment at 1310 nm,” Arch. Ophthalmol. 119, 1179–1185 (2001).
[PubMed]

Barroso, J.

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

Barton, J. K.

J. K. Barton, J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, and A. J. Welch, “Three-dimensional reconstruction of blood vessels from in vivo color Doppler optical coherence tomography images,” Dermatology 198, 355–361 (1999).
[CrossRef]

Bizheva, K. K.

Boppart, S. A.

J. S. Bredfeldt, C. Vinegoni, D. L. Marks, and S. A. Boppart, “Molecularly sensitive optical coherence tomography,” Opt. Lett. 30, 495–497 (2005).
[CrossRef] [PubMed]

A. L. Oldenburg, F. J.-J. Toublan, K. S. Suslick, A. Wei, and S. A. Boppart, “Magnetomotive contrast for in vivo optical coherence tomography,” Opt. Express 13, 6597–6614 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-17-6597.
[CrossRef] [PubMed]

D. L. Marks and S. A. Boppart, “Nonlinear Interferometric Vibrational Imaging,” Phys. Rev. Lett. 92, 1239051–1239054 (2004).
[CrossRef]

C. Vinegoni, J. S. Bredfeldt, D. L. Marks, and S. A. Boppart, “Nonlinear optical coherence enhancement for optical coherence tomography,” Opt. Express 12, 331–341 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-2-331.
[CrossRef] [PubMed]

C. Xu, D. L. Marks, M. N. Do, and S. A. Boppart, “Separation of absorption and scattering profiles in spectroscopic optical coherence tomography using a least-squares algorithm,” Opt. Express 12, 4790–4803 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4790.
[CrossRef] [PubMed]

T. M. Lee, A. L. Oldenburg, S. Sitafalwalla, D. L. Marks, W. Luo, F. J.-J. Toublan, K. S. Suslick, and S. A. Boppart, “Engineered Microsphere Contrast Agents for Optical Coherence Tomography,” Opt. Lett. 28, 1546–1548 (2003).
[CrossRef] [PubMed]

S. A. Boppart, B. E. Bouma, M. E. Brezinski, G. J. Tearney, and J. G. Fujimoto, “Imaging developing neural morphology using optical coherence tomography.,” J. Neurosci. Methods 70, 65–72 (1996), http://research.bmn.com/medline/search/results?uid=MDLN.97137646.
[CrossRef] [PubMed]

Bouma, B. E.

N. Nassif, B. Cense, B. H. Park, S. H. Yun, T. C. Chen, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography,” Opt. Lett. 29, 480–482 (2004).
[CrossRef] [PubMed]

S. A. Boppart, B. E. Bouma, M. E. Brezinski, G. J. Tearney, and J. G. Fujimoto, “Imaging developing neural morphology using optical coherence tomography.,” J. Neurosci. Methods 70, 65–72 (1996), http://research.bmn.com/medline/search/results?uid=MDLN.97137646.
[CrossRef] [PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics. (Academic: San Diego, CA, 1992).

Bredfeldt, J. S.

Brezinski, M. E.

S. A. Boppart, B. E. Bouma, M. E. Brezinski, G. J. Tearney, and J. G. Fujimoto, “Imaging developing neural morphology using optical coherence tomography.,” J. Neurosci. Methods 70, 65–72 (1996), http://research.bmn.com/medline/search/results?uid=MDLN.97137646.
[CrossRef] [PubMed]

Bruchez, M. P.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300, 1434╍1436 (2003).
[CrossRef] [PubMed]

Byer, R. G.

R. G. Byer, “Parametric oscillators and nonlinear materials'” In Nonlinear Optics, P. G. Harper and B. S. Wherrett, Eds. (Academic: London, 1977).

Campagnola, P. J.

W. Mohler, A. C. Millard, and P. J. Campagnola, “Second Harmonic Generation Imaging of Endogenous Structural Proteins,” Methods 29, 97–109 (2003).
[CrossRef] [PubMed]

Cang, H.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Celliers, P. M.

P. Stoller, P. M. Celliers, K. M. Reiser, and A. M. Rubenchik, “Quantitative Second-Harmonic Generation Microscopy in Collagen,” Appl. Optics 42, 5209 (2003).
[CrossRef]

Cense, B.

Chen, J.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Chen, T. C.

Chen, Z.

Choma, M. A.

Clark, S. W.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300, 1434╍1436 (2003).
[CrossRef] [PubMed]

Cobb, M. J.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Costela, A.

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

de Boer, J. F.

Delysse, S.

S. Delysse, J.-M. Nunzi, and C. Scala-Valero, “Picosecond optical Kerr ellipsometry determination of S1-Sn absorption spetra of xanthene dyes,” Appl. Phys. B 66, 439–444 (1998).
[CrossRef]

Do, M. N.

Drexler, W.

Faber, D. J.

Fercher, A. F.

Francisco, J. S.

J. I. Steinfeld, J. S. Francisco, and W. L. Hase, Chemical Kinetics and Dynamics. (Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1989).

Fujimoto, J. G.

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25, 111–113 (1999).
[CrossRef]

S. A. Boppart, B. E. Bouma, M. E. Brezinski, G. J. Tearney, and J. G. Fujimoto, “Imaging developing neural morphology using optical coherence tomography.,” J. Neurosci. Methods 70, 65–72 (1996), http://research.bmn.com/medline/search/results?uid=MDLN.97137646.
[CrossRef] [PubMed]

Garcia-Moreno, I.

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

Hamaguchi, H.-o.

H.-o. Hamaguchi, “Nonlinear Raman Spectroscopy'” In Nonlinear Spectroscopy for Molecular Structure Determination, R. W. Field, E. H. A. J. P. Maier, and S. Tsuchiya, Eds. (Blackwell Science, Ltd: Malden, MA, 1998).

Hase, W. L.

J. I. Steinfeld, J. S. Francisco, and W. L. Hase, Chemical Kinetics and Dynamics. (Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1989).

Heikal, A. A.

A. A. Heikal, S. T. Hess, and W. W. Webb, “Multiphoton molecular spectroscopy and excited-state dynamics of enhanced green fluorescent protein (EGFP): acid-base specificity,” Chem. Phys. 274, 37–55 (2001).
[CrossRef]

Hermann, B.

Hess, S. T.

A. A. Heikal, S. T. Hess, and W. W. Webb, “Multiphoton molecular spectroscopy and excited-state dynamics of enhanced green fluorescent protein (EGFP): acid-base specificity,” Chem. Phys. 274, 37–55 (2001).
[CrossRef]

Ippen, E. P.

Izatt, J. A.

B. E. Applegate, C. Yang, A. M. Rollins, and J. A. Izatt, “Polarization resolved second harmonic generation optical coherence tomography in collagen,” Opt. Lett. 29, 2252–2254 (2004).
[CrossRef] [PubMed]

C. Yang, L. E. L. McGuckin, J. D. Simon, M. A. Choma, B. E. Applegate, and J. A. Izatt, “Spectral triangulation molecular contrast optical coherence tomography with indocyanine green as the contrast agent,” Opt. Lett. 29, 2016–2018 (2004).
[CrossRef] [PubMed]

C. Yang, M. A. Choma, L. E. Lamb, J. D. Simon, and J. A. Izatt, “Protein-based molecular contrast optical coherence tomography with phytochrome as the contrast agent,” Opt. Lett. 29, 1396–1398 (2004).
[CrossRef] [PubMed]

M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, “Sensitivity advantage of swept-source and Fourier-domain optical coherence tomography,” Opt. Express 11, 2183–2189 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-18-2183.
[CrossRef] [PubMed]

K. D. Rao, M. A. Choma, S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “Molecular contrast in optical coherence tomography by use of a pump-probe technique,” Opt. Lett. 28, 340–342 (2003).
[CrossRef] [PubMed]

T. M. Yelbuz, M. A. Choma, L. Thrane, M. L. Kirby, and J. A. Izatt, “Optical coherence tomography - A new high-resolution imaging technology to study cardiac development in chick embryos,” Circulation 106, 2771–2774 (2002).
[CrossRef] [PubMed]

S. Radhakrishnan, A. M. Rollins, J. E. Roth, S. Yazdanfar, V. Westphal, D. S. Bardenstein, and J. A. Izatt, “Real-time optical coherence tomography of the anterior segment at 1310 nm,” Arch. Ophthalmol. 119, 1179–1185 (2001).
[PubMed]

J. K. Barton, J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, and A. J. Welch, “Three-dimensional reconstruction of blood vessels from in vivo color Doppler optical coherence tomography images,” Dermatology 198, 355–361 (1999).
[CrossRef]

B. E. Applegate and J. A. Izatt, Department of Biomedical Engineering, Duke University, Durham, NC 27708, are preparing a manuscript to be called “Transient Absorption and Lifetime Imaging with Ground State Recovery Pump-Probe Optical Coherence Tomography.”

Jiang, Y.

Kärtner, F. X.

Kimmey, M. B.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Kirby, M. L.

T. M. Yelbuz, M. A. Choma, L. Thrane, M. L. Kirby, and J. A. Izatt, “Optical coherence tomography - A new high-resolution imaging technology to study cardiac development in chick embryos,” Circulation 106, 2771–2774 (2002).
[CrossRef] [PubMed]

Kulkarni, M. D.

J. K. Barton, J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, and A. J. Welch, “Three-dimensional reconstruction of blood vessels from in vivo color Doppler optical coherence tomography images,” Dermatology 198, 355–361 (1999).
[CrossRef]

Kwant, G.

M. L. J. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light- absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

Lamb, L. E.

Landsman, M. L. J.

M. L. J. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light- absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

Larson, D. R.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300, 1434╍1436 (2003).
[CrossRef] [PubMed]

Lee, T. M.

Li, X. D.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25, 111–113 (1999).
[CrossRef]

Li, Z.-Y.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Luo, W.

Marks, D. L.

McGuckin, L. E. L.

Mik, E. G.

Millard, A. C.

W. Mohler, A. C. Millard, and P. J. Campagnola, “Second Harmonic Generation Imaging of Endogenous Structural Proteins,” Methods 29, 97–109 (2003).
[CrossRef] [PubMed]

Mohler, W.

W. Mohler, A. C. Millard, and P. J. Campagnola, “Second Harmonic Generation Imaging of Endogenous Structural Proteins,” Methods 29, 97–109 (2003).
[CrossRef] [PubMed]

Mook, G. A.

M. L. J. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light- absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

Morgner, U.

Nassif, N.

Nunzi, J.-M.

S. Delysse, J.-M. Nunzi, and C. Scala-Valero, “Picosecond optical Kerr ellipsometry determination of S1-Sn absorption spetra of xanthene dyes,” Appl. Phys. B 66, 439–444 (1998).
[CrossRef]

Oldenburg, A. L.

Owyoung, A.

A. Owyoung and P. S. Peercy, “Precise Characterixation of the Raman nonlinearity in benzene using nonlinear interferometry,” J. Appl. Physics 48, 674–677 (1977).
[CrossRef]

Park, B. H.

Peercy, P. S.

A. Owyoung and P. S. Peercy, “Precise Characterixation of the Raman nonlinearity in benzene using nonlinear interferometry,” J. Appl. Physics 48, 674–677 (1977).
[CrossRef]

Penzkofer, A.

S. Reindl and A. Penzkofer, “Triplet quantum yield determination by picosecond laser double-laser fluorescence excitation,” Chem. Phys. 213, 429–438 (1996).
[CrossRef]

Pitris, C.

Považay, B.

Radhakrishnan, S.

S. Radhakrishnan, A. M. Rollins, J. E. Roth, S. Yazdanfar, V. Westphal, D. S. Bardenstein, and J. A. Izatt, “Real-time optical coherence tomography of the anterior segment at 1310 nm,” Arch. Ophthalmol. 119, 1179–1185 (2001).
[PubMed]

Rao, K. D.

Rao, N.

Rao, V.

Reindl, S.

S. Reindl and A. Penzkofer, “Triplet quantum yield determination by picosecond laser double-laser fluorescence excitation,” Chem. Phys. 213, 429–438 (1996).
[CrossRef]

Reiser, K. M.

P. Stoller, P. M. Celliers, K. M. Reiser, and A. M. Rubenchik, “Quantitative Second-Harmonic Generation Microscopy in Collagen,” Appl. Optics 42, 5209 (2003).
[CrossRef]

Rollins, A. M.

Roth, J. E.

S. Radhakrishnan, A. M. Rollins, J. E. Roth, S. Yazdanfar, V. Westphal, D. S. Bardenstein, and J. A. Izatt, “Real-time optical coherence tomography of the anterior segment at 1310 nm,” Arch. Ophthalmol. 119, 1179–1185 (2001).
[PubMed]

Rubenchik, A. M.

P. Stoller, P. M. Celliers, K. M. Reiser, and A. M. Rubenchik, “Quantitative Second-Harmonic Generation Microscopy in Collagen,” Appl. Optics 42, 5209 (2003).
[CrossRef]

Saeki, F.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Sarunic, M. V.

Sastre, R.

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

Sattmann, H.

Scala-Valero, C.

S. Delysse, J.-M. Nunzi, and C. Scala-Valero, “Picosecond optical Kerr ellipsometry determination of S1-Sn absorption spetra of xanthene dyes,” Appl. Phys. B 66, 439–444 (1998).
[CrossRef]

Schmetterer, L.

Simon, J. D.

Sitafalwalla, S.

Srinivas, N.

Steinfeld, J. I.

J. I. Steinfeld, J. S. Francisco, and W. L. Hase, Chemical Kinetics and Dynamics. (Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1989).

Stoller, P.

P. Stoller, P. M. Celliers, K. M. Reiser, and A. M. Rubenchik, “Quantitative Second-Harmonic Generation Microscopy in Collagen,” Appl. Optics 42, 5209 (2003).
[CrossRef]

Suslick, K. S.

Tanielian, C.

C. Tanielian and C. Wolff, “Determination of the Parameters Controlling Singlet Oxygen Production via Oxygen and Heavy-Atom Enhancement of Triplet Yields,” J. Phys. Chem. 99, 9831–9837 (1995).
[CrossRef]

Tearney, G. J.

N. Nassif, B. Cense, B. H. Park, S. H. Yun, T. C. Chen, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography,” Opt. Lett. 29, 480–482 (2004).
[CrossRef] [PubMed]

S. A. Boppart, B. E. Bouma, M. E. Brezinski, G. J. Tearney, and J. G. Fujimoto, “Imaging developing neural morphology using optical coherence tomography.,” J. Neurosci. Methods 70, 65–72 (1996), http://research.bmn.com/medline/search/results?uid=MDLN.97137646.
[CrossRef] [PubMed]

Thrane, L.

T. M. Yelbuz, M. A. Choma, L. Thrane, M. L. Kirby, and J. A. Izatt, “Optical coherence tomography - A new high-resolution imaging technology to study cardiac development in chick embryos,” Circulation 106, 2771–2774 (2002).
[CrossRef] [PubMed]

Tian, P. F.

Tomov, I.

Toublan, F. J.-J.

Unterhuber, A.

van Leeuwen, T. G.

Vinegoni, C.

Wang, Y.

Warren, W. S.

Webb, W. W.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300, 1434╍1436 (2003).
[CrossRef] [PubMed]

A. A. Heikal, S. T. Hess, and W. W. Webb, “Multiphoton molecular spectroscopy and excited-state dynamics of enhanced green fluorescent protein (EGFP): acid-base specificity,” Chem. Phys. 274, 37–55 (2001).
[CrossRef]

Wei, A.

Welch, A. J.

J. K. Barton, J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, and A. J. Welch, “Three-dimensional reconstruction of blood vessels from in vivo color Doppler optical coherence tomography images,” Dermatology 198, 355–361 (1999).
[CrossRef]

Westphal, V.

S. Radhakrishnan, A. M. Rollins, J. E. Roth, S. Yazdanfar, V. Westphal, D. S. Bardenstein, and J. A. Izatt, “Real-time optical coherence tomography of the anterior segment at 1310 nm,” Arch. Ophthalmol. 119, 1179–1185 (2001).
[PubMed]

Wiley, B. J.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Williams, R. M.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300, 1434╍1436 (2003).
[CrossRef] [PubMed]

Wise, F. W.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300, 1434╍1436 (2003).
[CrossRef] [PubMed]

Wolff, C.

C. Tanielian and C. Wolff, “Determination of the Parameters Controlling Singlet Oxygen Production via Oxygen and Heavy-Atom Enhancement of Triplet Yields,” J. Phys. Chem. 99, 9831–9837 (1995).
[CrossRef]

Xia, Y.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Xu, C.

Yang, C.

Yazdanfar, S.

K. D. Rao, M. A. Choma, S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “Molecular contrast in optical coherence tomography by use of a pump-probe technique,” Opt. Lett. 28, 340–342 (2003).
[CrossRef] [PubMed]

S. Radhakrishnan, A. M. Rollins, J. E. Roth, S. Yazdanfar, V. Westphal, D. S. Bardenstein, and J. A. Izatt, “Real-time optical coherence tomography of the anterior segment at 1310 nm,” Arch. Ophthalmol. 119, 1179–1185 (2001).
[PubMed]

J. K. Barton, J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, and A. J. Welch, “Three-dimensional reconstruction of blood vessels from in vivo color Doppler optical coherence tomography images,” Dermatology 198, 355–361 (1999).
[CrossRef]

Yelbuz, T. M.

T. M. Yelbuz, M. A. Choma, L. Thrane, M. L. Kirby, and J. A. Izatt, “Optical coherence tomography - A new high-resolution imaging technology to study cardiac development in chick embryos,” Circulation 106, 2771–2774 (2002).
[CrossRef] [PubMed]

Yun, S. H.

Zhang, H.

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Zijlstra, W. G.

M. L. J. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light- absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

Zipfel, W. R.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300, 1434╍1436 (2003).
[CrossRef] [PubMed]

Appl. Optics (1)

P. Stoller, P. M. Celliers, K. M. Reiser, and A. M. Rubenchik, “Quantitative Second-Harmonic Generation Microscopy in Collagen,” Appl. Optics 42, 5209 (2003).
[CrossRef]

Appl. Phys. B (1)

S. Delysse, J.-M. Nunzi, and C. Scala-Valero, “Picosecond optical Kerr ellipsometry determination of S1-Sn absorption spetra of xanthene dyes,” Appl. Phys. B 66, 439–444 (1998).
[CrossRef]

Arch. Ophthalmol. (1)

S. Radhakrishnan, A. M. Rollins, J. E. Roth, S. Yazdanfar, V. Westphal, D. S. Bardenstein, and J. A. Izatt, “Real-time optical coherence tomography of the anterior segment at 1310 nm,” Arch. Ophthalmol. 119, 1179–1185 (2001).
[PubMed]

Chem. Phys. (3)

J. Barroso, A. Costela, I. Garcia-Moreno, and R. Sastre, “Wavelength dependence of the nonlinear absorption properties of laser dyes in solid and liquid solutions,” Chem. Phys. 238, 257–272 (1998).
[CrossRef]

S. Reindl and A. Penzkofer, “Triplet quantum yield determination by picosecond laser double-laser fluorescence excitation,” Chem. Phys. 213, 429–438 (1996).
[CrossRef]

A. A. Heikal, S. T. Hess, and W. W. Webb, “Multiphoton molecular spectroscopy and excited-state dynamics of enhanced green fluorescent protein (EGFP): acid-base specificity,” Chem. Phys. 274, 37–55 (2001).
[CrossRef]

Circulation (1)

T. M. Yelbuz, M. A. Choma, L. Thrane, M. L. Kirby, and J. A. Izatt, “Optical coherence tomography - A new high-resolution imaging technology to study cardiac development in chick embryos,” Circulation 106, 2771–2774 (2002).
[CrossRef] [PubMed]

Dermatology (1)

J. K. Barton, J. A. Izatt, M. D. Kulkarni, S. Yazdanfar, and A. J. Welch, “Three-dimensional reconstruction of blood vessels from in vivo color Doppler optical coherence tomography images,” Dermatology 198, 355–361 (1999).
[CrossRef]

J. Appl. Physics (1)

A. Owyoung and P. S. Peercy, “Precise Characterixation of the Raman nonlinearity in benzene using nonlinear interferometry,” J. Appl. Physics 48, 674–677 (1977).
[CrossRef]

J. Appl. Physiol. (1)

M. L. J. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light- absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol. 40, 575–583 (1976).
[PubMed]

J. Neurosci. Methods (1)

S. A. Boppart, B. E. Bouma, M. E. Brezinski, G. J. Tearney, and J. G. Fujimoto, “Imaging developing neural morphology using optical coherence tomography.,” J. Neurosci. Methods 70, 65–72 (1996), http://research.bmn.com/medline/search/results?uid=MDLN.97137646.
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (1)

J. Phys. Chem. (1)

C. Tanielian and C. Wolff, “Determination of the Parameters Controlling Singlet Oxygen Production via Oxygen and Heavy-Atom Enhancement of Triplet Yields,” J. Phys. Chem. 99, 9831–9837 (1995).
[CrossRef]

Methods (1)

W. Mohler, A. C. Millard, and P. J. Campagnola, “Second Harmonic Generation Imaging of Endogenous Structural Proteins,” Methods 29, 97–109 (2003).
[CrossRef] [PubMed]

Nano Lett. (1)

J. Chen, F. Saeki, B. J. Wiley, H. Cang, M. J. Cobb, Z.-Y. Li, L. Au, H. Zhang, M. B. Kimmey, X. D. Li, and Y. Xia, “Gold Nanocages: Bioconjugation and Their Potential Use as Optical Imaging Contrast Agents,” Nano Lett. 5, 473–477 (2005).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (12)

C. Yang, M. A. Choma, L. E. Lamb, J. D. Simon, and J. A. Izatt, “Protein-based molecular contrast optical coherence tomography with phytochrome as the contrast agent,” Opt. Lett. 29, 1396–1398 (2004).
[CrossRef] [PubMed]

T. M. Lee, A. L. Oldenburg, S. Sitafalwalla, D. L. Marks, W. Luo, F. J.-J. Toublan, K. S. Suslick, and S. A. Boppart, “Engineered Microsphere Contrast Agents for Optical Coherence Tomography,” Opt. Lett. 28, 1546–1548 (2003).
[CrossRef] [PubMed]

J. S. Bredfeldt, C. Vinegoni, D. L. Marks, and S. A. Boppart, “Molecularly sensitive optical coherence tomography,” Opt. Lett. 30, 495–497 (2005).
[CrossRef] [PubMed]

N. Nassif, B. Cense, B. H. Park, S. H. Yun, T. C. Chen, B. E. Bouma, G. J. Tearney, and J. F. de Boer, “In vivo human retinal imaging by ultrahigh-speed spectral domain optical coherence tomography,” Opt. Lett. 29, 480–482 (2004).
[CrossRef] [PubMed]

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25, 111–113 (1999).
[CrossRef]

D. J. Faber, E. G. Mik, M. C. G. Aalders, and T. G. van Leeuwen, “Light absorption of (oxy-)hemoglobin assessed by spectroscopic optical coherence tomography,” Opt. Lett. 28, 1436–1438 (2003).
[CrossRef] [PubMed]

D. J. Faber, E. G. Mik, M. C. G. Aalders, and T. G. van Leeuwen, “Toward assessment of blood oxygen saturation by spectroscopic optical coherence tomography,” Opt. Lett. 30, 1015–1017 (2005).
[CrossRef] [PubMed]

K. D. Rao, M. A. Choma, S. Yazdanfar, A. M. Rollins, and J. A. Izatt, “Molecular contrast in optical coherence tomography by use of a pump-probe technique,” Opt. Lett. 28, 340–342 (2003).
[CrossRef] [PubMed]

B. E. Applegate, C. Yang, A. M. Rollins, and J. A. Izatt, “Polarization resolved second harmonic generation optical coherence tomography in collagen,” Opt. Lett. 29, 2252–2254 (2004).
[CrossRef] [PubMed]

Y. Jiang, I. Tomov, Y. Wang, and Z. Chen, “Second-harmonic optical coherence tomography,” Opt. Lett. 29, 1090–1092 (2004).
[CrossRef] [PubMed]

P. F. Tian and W. S. Warren, “Ultrafast measurement of two-photon absorption by loss modulation,” Opt. Lett. 27, 1634╍1636 (2002).
[CrossRef]

C. Yang, L. E. L. McGuckin, J. D. Simon, M. A. Choma, B. E. Applegate, and J. A. Izatt, “Spectral triangulation molecular contrast optical coherence tomography with indocyanine green as the contrast agent,” Opt. Lett. 29, 2016–2018 (2004).
[CrossRef] [PubMed]

Photochem Photobiol (1)

C. Yang, “Molecular Contrast Optical Coherence Tomography: A Review,” Photochem Photobiol 81, 215–237 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

D. L. Marks and S. A. Boppart, “Nonlinear Interferometric Vibrational Imaging,” Phys. Rev. Lett. 92, 1239051–1239054 (2004).
[CrossRef]

Science (1)

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300, 1434╍1436 (2003).
[CrossRef] [PubMed]

Other (5)

H.-o. Hamaguchi, “Nonlinear Raman Spectroscopy'” In Nonlinear Spectroscopy for Molecular Structure Determination, R. W. Field, E. H. A. J. P. Maier, and S. Tsuchiya, Eds. (Blackwell Science, Ltd: Malden, MA, 1998).

J. I. Steinfeld, J. S. Francisco, and W. L. Hase, Chemical Kinetics and Dynamics. (Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1989).

B. E. Applegate and J. A. Izatt, Department of Biomedical Engineering, Duke University, Durham, NC 27708, are preparing a manuscript to be called “Transient Absorption and Lifetime Imaging with Ground State Recovery Pump-Probe Optical Coherence Tomography.”

R. G. Byer, “Parametric oscillators and nonlinear materials'” In Nonlinear Optics, P. G. Harper and B. S. Wherrett, Eds. (Academic: London, 1977).

R. W. Boyd, Nonlinear Optics. (Academic: San Diego, CA, 1992).

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Figures (2)

Fig. 1
Fig. 1

Plot of eqns (3) and (4) with units of power such that Ps =1, which implies that a ≤ 1.

Fig. 2.
Fig. 2.

Pump-probe schemes.

Tables (1)

Tables Icon

Table 1. Results of the derivations of expressions for PMCOCT described in the text.

Equations (38)

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S r = 2 P r P s 2 P r P s P 2 = 2 P r ( P s P s ( 1 aP s n 1 ) ½ )
2 P r ( a 2 P s n ½ ) .
SNR r ( 2 P r a 2 P s n ½ 2 P r ) 2 = a 2 P s 2 n 1 2
S nr = 2 P r aP s n
SNR nr ( 2 P r aP s n P r ) 2 = 4 aP s n .
S b = P s ( P s aP s n ) , and
S b = aP s n
SNR b ( aP s n 2 P s ) 2 = a 2 P s 2 n 1 2 ,
S d = aP s n , and
SNR d ( aP s n aP s n ) 2 = aP s n .
SNR b SNR d = aP s n 1 2 1 2 .
SNR OCT = ρR s Δ t 2 e P s ,
P SHOCT = aP ω 1 2
a = 2 χ SHG , eff ( 2 ) 2 πw 0 2 ω 1 2 n 1 2 n 2 c 3 ε 0 f 0 τ J 2 ,
b = 256 π 2 ω as 2 n p 2 n st n as c 4 r 2 χ CARS ( 3 ) N 1 2 l 2 [ sin c ( Δ kl 2 ) ] 2 ,
P NIVI = 256 π 2 ω as 2 n p 2 n st n as c 4 r 2 τ 2 f 0 2 χ CARS ( 3 ) N 1 2 l 2 [ sin c ( Δ kl 2 ) ] 2 P st P p 2 .
P TPA = ξp s 2 = 0.66 N 1 0 δn τf 0 hc arctan ( l n 2 z 0 ) P s 2
S = 2 P r P s ( 1 ½ ξP S ) .
S = 2 P r P s sin 2 ( ωt ) ( 1 ½ ξP S sin 2 ( ωt ) )
= 2 P r ( ( P S ½ ξP S 3 2 ) sin ( ωt ) + ξP S 3 2 sin ( 3 ωt ) ) .
S TPA = ξP S 3 2 .
P TPA OCT = ( ξ ) 2 P S 3 .
S ST = S ( λ a ) S ( λ c ) S ( λ b ) 1 = e ( σ a + σ c 2 σ b ) Nl 1 e σ b ( α 1 ) Nl 1
P SOCT = e 2 σ b Nl ( e σ b Nl ( α 1 ) 1 ) 2 9 ( ¼ e 2 μ s , a l + ¼ e 2 μ s , b l + 2 σ b Nl ( 1 α ) ) P s ,
S 1 = 2 P r P pr e σ 2 N 2 ( t ) l
S 2 = 2 P r P pr e σ 1 ( N 1 ( t ) N 2 ( t ) ) l ,
S 3 = 2 P r P pr e σ 3 q 2,3 N 2 ( t ) l
N 1 ( t ) = N 1 0 2 ρB + A [ A + ρB + ρBe ( 2 ρB + A ) t ] and
N 2 ( t ) = N 1 0 N 1 = ρBN 1 0 2 ρB + A [ 1 e ( 2 ρB + A ) t ] ,
ρB = σ 1 λ pu P pu hcπr 2 ,
N 1 ( τ ) = N 1 0 ( 1 σ 1 λ pu P pu hcπr 2 f 0 ) and
N 2 ( τ ) = N 1 0 σ 1 λ pu P pu hcπr 2 f 0 ,
S 1 = 2 P r P pr f 0 τ e σ 2 N 2 ( τ ) l 2 P r P pr f 0 τ ( 1 σ 2 l N 2 ( τ ) ) ,
S 1 = 2 P r P pr f 0 τ ( 1 σ 2 l N 1 0 σ 1 λ pu P pu hcπr 2 f 0 )
S 1 = 2 P r P pr ( 1 σ 2 l N 1 0 σ 1 λ pu P pu hcπr 2 f 0 ) .
S 1 ( ω ) = P r P pr ( σ 2 l N 1 0 σ 1 λ pu P pu 2 hcπr 2 f 0 ) and
P PPOCT = ( σ 2 l N 1 0 σ 1 λ pu P pu 2 hcπr 2 f 0 ) 2 P pr .
N md = N 2 10 SNR ( dB ) 10

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