Abstract

A large-aperture, electromagnetic model for coherent microscopy is presented and the inverse scattering problem is solved. Approximations to the model are developed for near-focus and far-from-focus operations. These approximations result in an image-reconstruction algorithm consistent with interferometric synthetic aperture microscopy (ISAM): this validates ISAM processing of optical-coherence-tomography and optical-coherence-microscopy data in a vectorial setting. Numerical simulations confirm that diffraction-limited resolution can be achieved outside the focal plane and that depth of focus is limited only by measurement noise and/or detector dynamic range. Furthermore, the model presented is suitable for the quantitative study of polarimetric coherent microscopy systems operating within the first Born approximation.

© 2007 Optical Society of America

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References

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2007 (3)

2006 (3)

2004 (2)

2003 (4)

2002 (2)

Z. Ding, Y. Zhao, H. Ren, J. S. Nelson, and Z. Chen, "Real-time phase-resolved optical coherence tomography and optical Doppler tomography," Opt. Express 10, 236-245 (2002).
[PubMed]

J. F. de Boer and T. E. Milner, "Review of polarization sensitive optical coherence tomography and Stokes vector determination," J. Biomed. Opt. 7, 359-371 (2002).
[CrossRef] [PubMed]

2000 (2)

1997 (1)

P. T. Gough and D. W. Hawkins, "Unified framework for modern synthetic aperture imaging algorithms," Int. J. Imaging Syst. Technol. 8, 343-358 (1997).
[CrossRef]

1994 (1)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

1990 (1)

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

1959 (2)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

1919 (1)

H. Weyl, "Expansion of electro magnetic waves on an even conductor," Ann. Phys. 60, 481-500 (1919).
[CrossRef]

Baleine, E.

Boppart, S. A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1980).

Carney, P. S.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Changhuei, Y.

M. Choma, M. Sarunic, Y. Changhuei, and J. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 111, 2183-2189 (2003).
[CrossRef]

Chen, Z.

Choma, M.

M. Choma, M. Sarunic, Y. Changhuei, and J. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 111, 2183-2189 (2003).
[CrossRef]

Chou, C.

Chou, Y. H.

Curlander, J. C.

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley-Interscience, 1991).

Davis, B. J.

de Boer, J. F.

Denk, W.

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

Ding, Z.

Dogariu, A.

Elder, J. B.

Feng, Y.

Fercher, A. F.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, "Optical coherence microscopy in scattering media," Opt. Lett. 19, 590-592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Gough, P. T.

P. T. Gough and D. W. Hawkins, "Unified framework for modern synthetic aperture imaging algorithms," Int. J. Imaging Syst. Technol. 8, 343-358 (1997).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Han, C. Y.

Hansen, P. C.

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems (SIAM, 1998).
[CrossRef]

Hariharan, P.

P. Hariharan, Optical Interferometry (Academic, 2003).

Hawkins, D. W.

P. T. Gough and D. W. Hawkins, "Unified framework for modern synthetic aperture imaging algorithms," Int. J. Imaging Syst. Technol. 8, 343-358 (1997).
[CrossRef]

Hee, M. R.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, "Optical coherence microscopy in scattering media," Opt. Lett. 19, 590-592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Hitzenberger, C. K.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Izatt, J.

M. Choma, M. Sarunic, Y. Changhuei, and J. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 111, 2183-2189 (2003).
[CrossRef]

Izatt, J. A.

Leitgeb, R.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Lyu, C. W.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1996), Chap. 3, pp. 92-146.

Marks, D. L.

McDonough, R. N.

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley-Interscience, 1991).

Milner, T. E.

J. F. de Boer and T. E. Milner, "Review of polarization sensitive optical coherence tomography and Stokes vector determination," J. Biomed. Opt. 7, 359-371 (2002).
[CrossRef] [PubMed]

Mujat, M.

Nelson, J. S.

Nussenzveig, H. M.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, 1992), Chap. 2.2, pp. 17-20.

Oldenburg, A. L.

Owen, G. M.

Peng, L. C.

Potton, R. J.

R. J. Potton, "Reciprocity in optics," Rep. Prog. Phys. 67, 717-754 (2004).
[CrossRef]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Ralston, T. S.

Ren, H.

Reynolds, J. J.

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Sarunic, M.

M. Choma, M. Sarunic, Y. Changhuei, and J. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 111, 2183-2189 (2003).
[CrossRef]

Saxer, C.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Sheppard, C. J. R.

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Strickler, J. H.

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

Swanson, E. A.

J. A. Izatt, M. R. Hee, G. M. Owen, E. A. Swanson, and J. G. Fujimoto, "Optical coherence microscopy in scattering media," Opt. Lett. 19, 590-592 (1994).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Tang, Y. H.

Wang, R. K.

Webb, W. W.

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

Weyl, H.

H. Weyl, "Expansion of electro magnetic waves on an even conductor," Ann. Phys. 60, 481-500 (1919).
[CrossRef]

Wiener, N.

N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (MIT, 1964).

Wilson, T.

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

Wolf, E.

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1996), Chap. 3, pp. 92-146.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1980).

Xiang, S.

Zhao, Y.

Ann. Phys. (1)

H. Weyl, "Expansion of electro magnetic waves on an even conductor," Ann. Phys. 60, 481-500 (1919).
[CrossRef]

Appl. Opt. (1)

Int. J. Imaging Syst. Technol. (1)

P. T. Gough and D. W. Hawkins, "Unified framework for modern synthetic aperture imaging algorithms," Int. J. Imaging Syst. Technol. 8, 343-358 (1997).
[CrossRef]

J. Biomed. Opt. (1)

J. F. de Boer and T. E. Milner, "Review of polarization sensitive optical coherence tomography and Stokes vector determination," J. Biomed. Opt. 7, 359-371 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (5)

Nat. Phys. (1)

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, "Interferometric synthetic aperture microscopy," Nat. Phys. 3, 129-134 (2007).
[CrossRef]

Opt. Express (3)

Opt. Lett. (5)

Proc. R. Soc. London, Ser. A (2)

E. Wolf, "Electromagnetic diffraction in optical systems. I. An integral representation of the image field," Proc. R. Soc. London, Ser. A 253, 349-357 (1959).
[CrossRef]

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London, Ser. A 253, 358-379 (1959).
[CrossRef]

Rep. Prog. Phys. (1)

R. J. Potton, "Reciprocity in optics," Rep. Prog. Phys. 67, 717-754 (2004).
[CrossRef]

Science (2)

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

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical coherence tomography," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Other (8)

J. C. Curlander and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley-Interscience, 1991).

T. Wilson and C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, 1984).

P. Hariharan, Optical Interferometry (Academic, 2003).

P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems (SIAM, 1998).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1980).

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, 1992), Chap. 2.2, pp. 17-20.

N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series (MIT, 1964).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1996), Chap. 3, pp. 92-146.

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Figures (11)

Fig. 1
Fig. 1

Basic illustration of a coherent microscope. A source feeds an interferometer where one arm produces a reference field and the other consists of illumination and detection from the sample to be imaged. The reference arm may contain an adjustable delay element (represented here by movable mirrors). In practical implementations, the Mach–Zehnder layout shown here is often replaced by a Michelson interferometer using a single objective lens. The sample is scanned mechanically or optically in two or three dimensions.

Fig. 2
Fig. 2

Diagram illustrating a single-lens OCT system. Some of the expressions derived in Section 2 are shown with the physical quantities they represent. Following standard practice, a ray optics description characterizes the lens. This description can then be interpreted as an angular spectrum and be used to calculate the fields in the vicinity of the focal spot.

Fig. 3
Fig. 3

Illustration of the Fourier-domain relation between the collected data and the object. A point ( Q , k ) in the data corresponds to the point Q = ( Q , 2 k z ( Q 2 ) ) in the Fourier-domain representation of the object. Thus the two-dimensional Fourier transform of the data at wavenumber k gives the object’s three-dimensional Fourier components at the same lateral frequencies and at a distance of 2 k from the origin.

Fig. 4
Fig. 4

Reconstructed images for point scatterers lying on the z axis. Images (a), (c), and (e) show standard reconstructions, while (b), (d), and (f) show ISAM resampling-based reconstructions. Images (a) and (b) correspond to a scatterer at ( 0 , 0 , 1 ) μ m , (c) and (d) are for a scatterer at ( 0 , 0 , 2 ) μ m , and (e) and (f) are for a scatterer at position ( 0 , 0 , 5 ) μ m . The two-dimensional plots shown are a lateral-axial slice of the respective three-dimensional reconstructions. The images are plotted in normalized units, where the peak value of (a) is 1. Note the drop in signal as the z position of the scatterer increases.

Fig. 5
Fig. 5

Real part of the Fourier-domain representations of the reconstructions from Fig. 4. The standard OCT reconstructions, shown on the left, stretch the Fourier representation of the data by a factor of 2 axially and flip the axial Fourier axis. The ISAM resampling approach (results shown on the right) can be seen to correct the data so as to better match the expected Fourier spectra of the object. In this case the Fourier-domain objects are complex exponentials oscillating in the axial direction—i.e., the oscillation crests should be straight.

Fig. 6
Fig. 6

Integral of the intensity falling on the detector plotted for a single scatterer as a function of its axial position. Several NAs are considered, and the intensity is calculated for Rayleigh ranges of 0.001 to 50 with 25 logarithmically spaced points. An × marks the 1 Rayleigh range point for each plot.

Fig. 7
Fig. 7

Noise-free reconstructions of an object consisting of ten point scatterers positioned in the x z plane at [(5.5,0,0), (0,0,1), ( 4.5 , 0 , 4.5 ) , (0,0,5), ( 2 , 0 , 7 ) , ( 2 , 0 , 15 ) , ( 1 , 0 , 15 ) , ( 2 , 0 , 16 ) , (12,0,17), and ( 20 , 0 , 25 ) ] μ m . The x z plane of the three-dimensional reconstructions are shown. Reconstructions for the standard OCT method are shown in (a), (c), and (e), while ISAM Fourier-resampling reconstructions are shown in (b), (d), and (f). Both methods include the axial gain function to boost out-of-focus planes. The NA used is 0.2 in (a) and (b), 0.4 in (c) and (d), and 0.75 in (e) and (f). The image scale is normalized to the maximum reconstruction value for the 0.2 NA data.

Fig. 8
Fig. 8

Noisy reconstructions of the same object considered in Fig. 7 using the same instrument parameters. The noise level considered results in a SNR of 0 dB in the 0.2-NA data. OCT reconstructions are shown on the left and ISAM reconstructions on the right.

Fig. 9
Fig. 9

Noisy reconstructions of the same object considered in Fig. 7, using the same instrument parameters. The noise level considered results in a SNR of 10 dB in the 0.2-NA data. OCT reconstructions are shown on the left and ISAM reconstructions on the right.

Fig. 10
Fig. 10

Noise-free reconstructions from a 0.05-NA system imaging an object with point scatterers in the x z plane at positions of [(22,0,0), (0,0,30), ( 18 , 0 , 135 ) , (0,0,150), ( 8 , 0 , 210 ) , ( 8 , 0 , 450 ) , ( 4 , 0 , 450 ) , ( 8 , 0 , 495 ) , and (48,0,510)] μ m . The x z plane of the reconstructions is shown in (a) and (b), along with the x z detail in (c) and (d) corresponding to the dashed square, and the x y detail in (e) and (f) from the plane marked with a broken line. Images for the standard OCT method are shown on the left and for the ISAM Fourier-resampling algorithm on the right. Both reconstructions include the axial gain function to boost out-of-focus planes. The image scale is normalized to the maximum reconstruction value.

Fig. 11
Fig. 11

Noise-free reconstructions from a 0.1-NA system imaging the object described in Fig. 10. The image scale is normalized to the maximum reconstruction value for the 0.05-NA data. The OCT reconstruction is shown on the left and the ISAM reconstruction on the right.

Equations (60)

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I ( r ( o ) ; k ) = E ( r ) ( k ) + E ( s ) ( r ( o ) ; k ) 2 = E ( r ) ( k ) 2 + E ( s ) ( r ( o ) ; k ) 2 + 2 Re { [ E ( r ) ( k ) ] H E ( s ) ( r ( o ) ; k ) } ,
S ( r ( o ) ; k ) = [ E ( r ) ( k ) ] H E ( s ) ( r ( o ) ; k ) .
S T ( r ( o ) ; τ ) = 1 2 π S ( r ( o ) ; k ( ω ) ) e i ω τ d ω .
E ( l ) ( σ x , σ y ) = A ¯ ( σ x , σ y ) E ( i ) P ( k ) .
σ z ( σ x , σ y ) = + 1 σ x 2 σ y 2 .
g ( r r ( o ) ; k ) = i k 2 π Ω A ¯ ( σ x , σ y ) E ( i ) σ z ( σ x , σ y ) e i k σ ( r r ( o ) ) d σ x d σ y .
E ( u ) ( r , r ( o ) ; k ) = k 2 P ( k ) G ¯ ( r , r ; k ) η ¯ ( r ) g ( r r ( o ) ; k ) d 3 r .
G ¯ ( r , r ; k ) = i k 2 π Ω D ¯ ( σ x , σ y ) σ z ( σ x , σ y ) e ± i k σ ( r r ) d σ x d σ y , z z .
E ( u ) ( r , r ( o ) ; k ) = k 2 P ( k ) i k 2 π Ω D ¯ ( σ x , σ y ) σ z ( σ x , σ y ) e i k σ ( r r ) d σ x d σ y η ¯ ( r ) g ( r r ( o ) ; k ) d 3 r = k 2 P ( k ) i k 2 π Ω D ¯ ( σ x , σ y ) σ z ( σ x , σ y ) e i k σ ( r r ( o ) ) e i k σ ( r r ( o ) ) d σ x d σ y η ¯ ( r ) g ( r r ( o ) ; k ) d 3 r .
S ( r ( o ) , k ) = k 2 μ r P ( k ) 2 ( E ( d ) ) H Ω B ¯ ( σ x , σ y ) i k 2 π D ¯ ( σ x , σ y ) σ z ( σ x , σ y ) e i k σ ( r r ( o ) ) d σ x d σ y η ¯ ( r ) g ( r r ( o ) ; k ) d 3 r .
B ¯ ( σ x , σ y ) D ¯ ( σ x , σ y ) = B ¯ ( σ x , σ y ) .
f ( r r ( o ) ; k ) = i k 2 π Ω B ¯ T ( σ x , σ y ) ( E ( d ) ) * σ z ( σ x , σ y ) e i k σ ( r r ( o ) ) d σ x d σ y ,
S ( r ( o ) , k ) = k 2 μ r P ( k ) 2 f T ( r r ( o ) ; k ) η ¯ ( r ) g ( r r ( o ) ; k ) d 3 r = k 2 μ r P ( k ) 2 f α ( r r ( o ) ; k ) g β ( r r ( o ) ; k ) η α β ( r ) d 3 r = h α β ( r ( o ) r ; k ) η α β ( r ) d 3 r .
h α β ( r ; k ) = μ r k 2 P ( k ) 2 f α ( r ; k ) g β ( r ; k ) .
B ¯ T ( σ x , σ y ) = A ¯ ( σ x , σ y ) .
E ( d ) = [ E ( i ) ] * .
S ( r ( o ) , k ) = k 2 μ r P ( k ) 2 g T ( r r ( o ) ; k ) η ¯ ( r ) g ( r r ( o ) ; k ) d 3 r .
S ( r ( o ) , k ) = k 2 μ r P ( k ) 2 g T ( r r ( o ) ; k ) [ ] d 3 r .
r ( o ) = ( x ( o ) , y ( o ) , z ( o ) ) = ( r ( o ) , z ( o ) ) .
S ( r ( o ) , k ) = h α β ( r ( o ) r , z ( o ) z ; k ) η α β ( r , z ) d 2 r d z .
S ̃ ( Q , k ) = h ̃ α β ( Q , z ( o ) z ; k ) η ̃ α β ( Q , z ) d z .
f α ( r ; k ) = i k 2 π Ω F α ( σ x , σ y ) σ z ( σ x , σ y ) e i k σ r d σ x d σ y ,
g β ( r ; k ) = i k 2 π Ω G β ( σ x , σ y ) σ z ( σ x , σ y ) e i k σ r d σ x d σ y ,
F α ( σ x , σ y ) = [ B ¯ T ( σ x , σ y ) ( E ( d ) ) * ] α ,
G β ( σ x , σ y ) = [ A ¯ ( σ x , σ y ) E ( i ) ] β .
f ̃ α ( Q , z ; k ) = 2 π i F α ( Q k ) k z ( Q ) e i k z ( Q ) z ,
g ̃ β ( Q , z ; k ) = 2 π i G β ( Q k ) k z ( Q ) e i k z ( Q ) z ,
h ̃ α β ( Q , z ; k ) = k 2 μ r P ( k ) 2 [ f ̃ α ( Q , z ; k ) g ̃ β ( Q , z ; k ) ] .
h ̃ α β ( Q , z ; k ) = 4 π 2 k 2 μ r P ( k ) 2 F α ( q k ) k z ( q ) G β ( Q q k ) k z ( Q q ) e i [ k z ( q ) + k z ( Q q ) ] z d 2 q .
q ( stat. ) = Q 2 .
h ̃ α β ( Q , z ; k ) i 4 π 3 k z μ r P ( k ) 2 e i 2 k z ( Q 2 ) z F α ( Q 2 k ) G β ( Q 2 k ) .
F α ( σ x , σ y ) = F ̌ α ( σ x , σ y ) σ z ( σ x , σ y ) ,
G β ( σ x , σ y ) = G ̌ β ( σ x , σ y ) σ z ( σ x , σ y ) .
h ̃ α β ( Q , z ; k ) = 4 π 2 k μ r P ( k ) 2 F ̌ α ( q k ) k z ( q ) G ̌ β ( Q q k ) k z ( Q q ) e i [ k z ( q ) + k z ( Q q ) ] z d 2 q .
e i [ k z ( q ) + k z ( Q q ) ] z k z ( q ) k z ( Q q ) = l = 0 m = 0 κ ( l , m , q ( p ) ; k ) ( q x q x ( p ) ) l ( q y q y ( p ) ) m ,
κ ( l , m , q ( p ) ; k ) = ( l + m ) l q x m q y e i [ k z ( q ) + k z ( Q q ) ] z k z ( q ) k z ( Q q ) q = q ( p ) .
h ̃ α β ( Q , z ; k ) = 4 π 2 k μ r P ( k ) 2 l = 0 m = 0 κ ( l , m , q ( p ) ; k ) F ̌ α ( q k ) G ̌ β ( Q q k ) ( q x q x ( p ) ) l ( q y q y ( p ) ) m d 2 q .
h ̃ α β ( Q , z ; k ) 4 π 2 k μ r P ( k ) 2 e i [ k z ( q ( p ) ) + k z ( Q q ( p ) ) ] z k z ( q ( p ) ) k z ( Q q ( p ) ) F ̌ α ( q k ) G ̌ β ( Q q k ) d 2 q .
h ̃ α β ( Q , z ; k ) 4 π 2 k μ r P ( k ) 2 K α β ( Q ; k ) k z ( Q 2 ) e i 2 k z ( Q 2 ) z ,
K α β ( Q ; k ) = F ̌ α ( q k ) G ̌ β ( Q q k ) d 2 q .
h ̃ α β ( Q , z ; k ) H α β ( Q ; k ) ρ ( z ) e i 2 k z ( Q 2 ) z ,
H α β ( Q ; k ) = { H α β ( N ) ( Q ; k ) = 4 π 2 k μ r P ( k ) 2 K α β ( Q ; k ) k z ( Q 2 ) z 1 k N A 2 H α β ( F ) ( Q ; k ) = i 4 π 3 k z ( Q 2 ) μ r P ( k ) 2 F ̌ α ( Q 2 k ) G ̌ β ( Q 2 k ) z 1 k N A 2 } ,
ρ ( z ) = { ρ ( N ) ( z ) = 1 z 1 k N A 2 ρ ( F ) ( z ) = 1 z z 1 k N A 2 } .
e γ 2 ( x 2 + y 2 ) ( 2 ζ 2 ) ,
G ̌ ( σ x , σ y ) = e γ 2 ( σ x 2 + σ y 2 ) 2 .
γ = 2 N A .
F ̌ ( σ x , σ y ) = e γ 2 ( σ x 2 + σ y 2 ) 2 .
K ( Q ; k ) = e γ 2 ( q x 2 + q y 2 ) ( 2 k 2 ) e γ 2 ( ( Q x q x ) 2 + ( Q y q y ) 2 ) ( 2 k 2 ) d q = e γ 2 ( Q x 2 + Q y 2 ) ( 4 k 2 ) e γ 2 ( ( q x Q x 2 ) 2 + ( q y Q y 2 ) 2 ) k 2 d q = ( e γ 2 { [ Q x ( 2 k ) ] 2 + [ Q y ( 2 k ) ] 2 } 2 2 ) 2 π k 2 γ 2 = π k 2 γ 2 F ̌ ( Q 2 k ) G ̌ ( Q 2 k ) .
H ( N ) ( Q ; k ) = 4 π 3 k 3 μ r P ( k ) 2 1 γ 2 k z ( Q 2 ) F ̌ ( Q 2 k ) G ̌ ( Q 2 k ) ,
H ( F ) ( Q ; k ) = i 4 π 3 k z ( Q 2 ) μ r P ( k ) 2 F ̌ ( Q 2 k ) G ̌ ( Q 2 k ) .
h ̃ ( Q , z ; k ) = [ 1 H ( N ) ( Q ; k ) ρ ( N ) ( z ) + 1 H ( F ) ( Q ; k ) ρ ( F ) ( z ) ] 1 e i 2 k z ( Q 2 ) z .
z = γ 2 k z 2 ( Q 2 ) k 3 γ 2 k = 2 k N A 2 = λ π N A 2 .
S ̃ ( Q , k ) = H α β ( Q ; k ) ρ ( z ) η ̃ α β ( Q , z ) e i 2 k z ( Q 2 ) z d z .
η ¯ ( r ) = ρ ( z ) η ¯ ( r ) .
S ̃ ( Q , k ) = H α β ( Q ; k ) η ̃ α β ( Q , z ) e i 2 k z ( Q 2 ) z d z .
S ̃ ( Q , k ) = H α β ( Q ; k ) η ͌ α β ( Q , 2 k z ( Q 2 ) ) .
ϱ ( z ) = { 1 z < z ( c ) z ( c ) z z z ( c ) } .
η ¯ ( r ) = η ( r ) [ 1 0 0 0 1 0 0 0 1 ] .
S ̃ ( Q ; k ) = ( α H α α ( Q ; k ) ) η ͌ ( Q , 2 k z ( Q 2 ) ) ,
η ( r ) = ρ ( z ) η ( r ) .

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