Abstract

The calculation of angular light-scattering distributions is considered for low-coherence light incident on a polydisperse particle size distribution of scatterers. As low-coherence light is now commonly used in interferometry schemes when applied to biomedical imaging, the difference between detecting scattered intensity and interferometrically detecting the scattered field is examined. An expression is derived that allows the presence of multiple wavelengths λ and particle sizes d to be described by a single distribution in the size parameter x=πd/λ, which simplifies numerical calculations. The applicability of this expression is examined numerically.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [PubMed]
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A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, M. S. Feld, “In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry,” Cancer Res. 63, 3556–3559 (2003).
[PubMed]

J. W. Pyhtila, R. N. Graf, A. Wax, “Determining nuclear morphology using an improved angle-resolved low coherence interferometry system,” Opt. Express 11, 3473–3484 (2003); www.opticsexpress.org .
[CrossRef] [PubMed]

2002

A. Wax, C. Yang, V. Backman, M. Kalashnikov, R. R. Dasari, M. S. Feld, “Determination of particle size using the angular distribution of backscattered light as measured with low-coherence interferometry,” J. Opt. Soc. Am. A 19, 737–744 (2002).
[CrossRef]

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, M. S. Feld, “Cell organization and sub-structure measured using angle-resolved low coherence interferometry,” Biophys. J. 82, 2256–2264 (2002).
[CrossRef] [PubMed]

2000

1998

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

A. Dogariu, E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23, 1340–1342 (1998).
[CrossRef]

1996

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

1995

P. Gorner, D. Bemer, J. F. Fabries, “Photometer measurement of polydisperse aerosols,” J. Aerosol Sci. 26, 1281–1302 (1995).
[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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

D. L. Hofeldt, R. K. Hanson, “Instantaneous imaging of particle size and spatial distribution in two-phase flows,” Appl. Opt. 30, 4936–4949 (1991).
[CrossRef] [PubMed]

1984

1979

1967

Backman, V.

A. Wax, C. Yang, V. Backman, M. Kalashnikov, R. R. Dasari, M. S. Feld, “Determination of particle size using the angular distribution of backscattered light as measured with low-coherence interferometry,” J. Opt. Soc. Am. A 19, 737–744 (2002).
[CrossRef]

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, M. S. Feld, “Cell organization and sub-structure measured using angle-resolved low coherence interferometry,” Biophys. J. 82, 2256–2264 (2002).
[CrossRef] [PubMed]

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Badizadegan, K.

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, M. S. Feld, “Cell organization and sub-structure measured using angle-resolved low coherence interferometry,” Biophys. J. 82, 2256–2264 (2002).
[CrossRef] [PubMed]

Barber, P. W.

Bemer, D.

P. Gorner, D. Bemer, J. F. Fabries, “Photometer measurement of polydisperse aerosols,” J. Aerosol Sci. 26, 1281–1302 (1995).
[CrossRef]

Boone, C. W.

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, M. S. Feld, “In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry,” Cancer Res. 63, 3556–3559 (2003).
[PubMed]

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, M. S. Feld, “Cell organization and sub-structure measured using angle-resolved low coherence interferometry,” Biophys. J. 82, 2256–2264 (2002).
[CrossRef] [PubMed]

Brodsky, A.

Brodsky, A. M.

Burgess, L. W.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Chen, H. C. H.

Conwell, P. R.

Crawford, J. M.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Dasari, R. R.

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, M. S. Feld, “In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry,” Cancer Res. 63, 3556–3559 (2003).
[PubMed]

A. Wax, C. Yang, V. Backman, M. Kalashnikov, R. R. Dasari, M. S. Feld, “Determination of particle size using the angular distribution of backscattered light as measured with low-coherence interferometry,” J. Opt. Soc. Am. A 19, 737–744 (2002).
[CrossRef]

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, M. S. Feld, “Cell organization and sub-structure measured using angle-resolved low coherence interferometry,” Biophys. J. 82, 2256–2264 (2002).
[CrossRef] [PubMed]

Dogariu, A.

Draper, J. S.

Fabries, J. F.

P. Gorner, D. Bemer, J. F. Fabries, “Photometer measurement of polydisperse aerosols,” J. Aerosol Sci. 26, 1281–1302 (1995).
[CrossRef]

Feld, M. S.

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, M. S. Feld, “In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry,” Cancer Res. 63, 3556–3559 (2003).
[PubMed]

A. Wax, C. Yang, V. Backman, M. Kalashnikov, R. R. Dasari, M. S. Feld, “Determination of particle size using the angular distribution of backscattered light as measured with low-coherence interferometry,” J. Opt. Soc. Am. A 19, 737–744 (2002).
[CrossRef]

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, M. S. Feld, “Cell organization and sub-structure measured using angle-resolved low coherence interferometry,” Biophys. J. 82, 2256–2264 (2002).
[CrossRef] [PubMed]

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

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, 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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Gorner, P.

P. Gorner, D. Bemer, J. F. Fabries, “Photometer measurement of polydisperse aerosols,” J. Aerosol Sci. 26, 1281–1302 (1995).
[CrossRef]

Graf, R. N.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hamano, T.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Hanson, R. K.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hofeldt, D. L.

Holland, A. C.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Itzkan, I.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Izatt, J. A.

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Kalashnikov, M.

Kobayashi, K.

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Kulkarni, M. D.

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Lima, C.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Manoharan, R.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Muller, M. G.

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, M. S. Feld, “In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry,” Cancer Res. 63, 3556–3559 (2003).
[PubMed]

Nines, R.

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, M. S. Feld, “In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry,” Cancer Res. 63, 3556–3559 (2003).
[PubMed]

Nusrat, A.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Perelman, L. T.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Popescu, G.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Pyhtila, J. W.

Rushforth, C. K.

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Seiler, M.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Shields, S.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Sivak, M. V.

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Steele, V. E.

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, M. S. Feld, “In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry,” Cancer Res. 63, 3556–3559 (2003).
[PubMed]

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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Stoner, G. D.

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, M. S. Feld, “In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry,” Cancer Res. 63, 3556–3559 (2003).
[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, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Thurber, S. R.

Van Dam, J.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Wallace, M.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Wang, D. S.

Wang, H. W.

J. A. Izatt, M. D. Kulkarni, H. W. Wang, K. Kobayashi, M. V. Sivak, “Optical coherence tomography and microscopy in gastrointestinal tissues,” IEEE J. Sel. Top. Quantum Electron. 2, 1017–1028 (1996).
[CrossRef]

Wax, A.

J. W. Pyhtila, R. N. Graf, A. Wax, “Determining nuclear morphology using an improved angle-resolved low coherence interferometry system,” Opt. Express 11, 3473–3484 (2003); www.opticsexpress.org .
[CrossRef] [PubMed]

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, M. S. Feld, “In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry,” Cancer Res. 63, 3556–3559 (2003).
[PubMed]

A. Wax, C. Yang, V. Backman, M. Kalashnikov, R. R. Dasari, M. S. Feld, “Determination of particle size using the angular distribution of backscattered light as measured with low-coherence interferometry,” J. Opt. Soc. Am. A 19, 737–744 (2002).
[CrossRef]

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, M. S. Feld, “Cell organization and sub-structure measured using angle-resolved low coherence interferometry,” Biophys. J. 82, 2256–2264 (2002).
[CrossRef] [PubMed]

Wolf, E.

Wyatt, P. J.

Yang, C.

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, M. S. Feld, “Cell organization and sub-structure measured using angle-resolved low coherence interferometry,” Biophys. J. 82, 2256–2264 (2002).
[CrossRef] [PubMed]

A. Wax, C. Yang, V. Backman, M. Kalashnikov, R. R. Dasari, M. S. Feld, “Determination of particle size using the angular distribution of backscattered light as measured with low-coherence interferometry,” J. Opt. Soc. Am. A 19, 737–744 (2002).
[CrossRef]

Yang, C. H.

A. Wax, C. H. Yang, M. G. Muller, R. Nines, C. W. Boone, V. E. Steele, G. D. Stoner, R. R. Dasari, M. S. Feld, “In situ detection of neoplastic transformation and chemopreventive effects in rat esophagus epithelium using angle-resolved low-coherence interferometry,” Cancer Res. 63, 3556–3559 (2003).
[PubMed]

Zonios, G.

L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627–631 (1998).
[CrossRef]

Appl. Opt.

Appl. Spectrosc.

Biophys. J.

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Figures (2)

Fig. 1
Fig. 1

Contour plot of the weighting function used to evaluate the total-scattering amplitude given in Eq. (26). The variables k and D are normalized by the corresponding mean values, ko and Do. Lines of constant size parameter x are indicated as dashed lines, and the linear approximation is shown as a dotted line.

Fig. 2
Fig. 2

Plot of numerically calculated size distribution (points) and best-fit Gaussian distribution (solid curve). The width of Gaussian distribution, Δx=(0.22+0.32)1/2=0.361 is based on the approximate model derived above, with parameters Δk=0.2ko and ΔD=0.3Do.

Equations (35)

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Es(kf)= Eo(k)S(kf, k)d3k,
Eo(k)= Uok(k)dk,
Eo(k)= Uok(k)dk= Uokδ2(k)δ(kz-k)dk,
Es(kf)= Uokδ2(k)δ(kz-k)S(kf, k)d3kdkz= UokS(kf, kzˆ)dk.
Is(kf)|Es(kf)|2=UokUok*S(kf, kzˆ)S*(kf, kzˆ)dkdk.
UokUok*=δk,k exp[-(k-ko)2/2Δk2];
Is(kf)|Es(kf)|2=exp[-(k-ko)2/2Δk2]|S(kf, kzˆ)|2dk.
Er(k, q, Δl)= Uokδ2(k-q)δ(kz-k)×exp(ikzΔl)dk,
Ib(q, Δl)EsEr*+c.c.=UokUok*δ2(k-q)δ(kz-k)×exp(ikzΔl)S(kf, kzˆ)dkdk+c.c.,
Ib(q, Δl)EsEr*+c.c.=exp[-(k-ko)2/2Δk2]S(q, kzˆ)×exp(ikΔl)dk+c.c.,
Ib(q, Δl)EsEr*+c.c.=2exp[-(k-ko)2/2Δk2]|S(q, kzˆ)|×cos(kΔl+ϕsk)dk,
Ib(q, Δl)EsEr*+c.c.=2exp[-(k-ko)2/2Δk2]|S(q, kzˆ)|×cos(kΔl+δt+ϕsk)dk,
|Ib(q)|2t=exp[-(k-ko)2/Δk2]|S(q, kzˆ)|2dk,
Stot(kf, k)=n Sn(kf, k)exp[ir(kf-k)],
Usk(kf)=UokStot(kf, k=kzˆ)=Uokn Sn(kf, k=kzˆ)exp[ir(kf-k)].
Is(kf)|Usk(kf)|2=|Uok|2n,m |Sn(kf, kzˆ)||Sm(kf, kzˆ)×|2 cos[(rn-rm)(kf-kzˆ)+ϕnmk],
Is(kf)|Uok|2n=m |Sn(kf, kzˆ)|2+|Uok|2nm |Sn(kf, kzˆ)||Sm(kf, kzˆ)|×2 cos[(rn-rm)(kf-kzˆ)+ϕnmk].
Urk(k, q, Δl)=Uokδ2(k-q)δ(kz-k)exp(ikzΔl),
Ibk(q, Δl)UskUrk*+c.c.=UokUok*n Sn(q, kzˆ)×cos[kΔl+δt+r(kf-k)]
|Ibk(q, Δl)|2tUokUok*2n,m Sn(q, kzˆ)Sm*(q, kzˆ)×cos[kΔl+δt+rn(kf-k)]×cos[kΔl+δt+rm(kf-k)]t.
|Ibk(q, Δl)|2t|Uok|2n=m |Sn(q, kzˆ)|2+|Uok|2nm Sn(q, kzˆ)Sm*(q, kzˆ)×cos(kΔl+δt+rn(kf-k))×cos(kΔl+δt+rm(kf-k))t.
|Uok|4n=m |Sn(q, kzˆ)|2|Uok|4  n(D)|SD(q, kzˆ)|2dD,
Is(kf)|Es|2=exp[-(k-ko)2/2Δk2]n=m |Sn(kf, kzˆ)|2dk,
Is(kf)|Es|2=exp[-(k-ko)2/2Δk2]×n(D)|SD(kf, kzˆ)|2dDdk,
|Ib(q, Δl=0)|2t=exp[-(k-ko)2/Δk2]×n(D)|SD(q, kzˆ)|2dDdk,
|Ib(q, Δl=0)|2t=exp[-(k-ko)2/Δk2]×exp[-(D-Do)2/ΔD2]|×SD(q, kzˆ)|2dDdk.
|Ib(q)|2t= w(k, D)|SD(q, kzˆ)|2dDdk,
w(k, D)=exp[-k2/Δk2]exp[-D2/ΔD2],
k=k/ko-1,Δk=Δk/ko,D=D/Do-1,ΔD=ΔD/Do.
w(x, y)=exp[-(x/2+y)2/Δk2]×exp[-(x/2-y)2/ΔD2].
w(x, y)=exp[-(x/2)2(1/Δk2+1/ΔD2)]×exp[-(y)2/(1/Δk2+1/ΔD2)]×exp[-xy(1/Δk2-1/ΔD2)],
|Ib(θ)|2t=1koDo  w(x, y)|SX(θ)|2dxdy,
|Ib(θ)|2t=1koDo  |SX(θ)|2×exp[-x2/(Δk2+ΔD2)]dx.
|Ib(θ)|2t=|SX(θ)|2 exp[-(x-xo)2/Δx2]dx,
xo=koDoandΔx2/xo2=Δk2+ΔD2=Δk2/ko2+ΔD2/Do2.

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