Abstract

Simulation techniques are developed for high-numerical-aperture (NA) polarized microscopy with Babinet’s principle, including partial coherence and vector diffraction for non-periodic geometries. The model includes vector illumination and diffraction in high-NA (up to NA=3.5) object space that is imaged into low-NA image space and recorded on an image sensor. A mathematical model for the Babinet approach is developed and interpreted that includes partial coherence using expanded mutual intensity, where object reflective characteristics modify the coherence functions. Simulation results of the Babinet’s principle approach are compared with those of rigorous coupled wave theory (RCWT) for periodic structures to investigate the accuracy of this approach and its limitations.

© 2010 Optical Society of America

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  1. G. S. Kino and S. M. Mansfield, “Solid immersion lens photon tunneling microscope,” (invited paper), Proc. SPIE  1556, 2–10(1992).
    [CrossRef]
  2. T. Chen, T. D. Milster, S.-H. Yang, and D. Hansen, “Evanescent imaging with induced polarization by using a solid immersion lens,” Opt. Lett.  32, 124–126, January 2007.
    [CrossRef]
  3. T. Chen, T. D. Milster, and S.-H. Yang, “Experimental investigation of photomask with near-field polarization imaging,” Proc. SPIE  6349, 634953 (2006).
    [CrossRef]
  4. R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE  5567, 887–893(2004).
    [CrossRef]
  5. K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, S. Masuhara, M. Furuki, and M. Yamamoto, “Readout Method for Read Only Memory Signal and Air Gap Control Signal in a Near Field Optical Disc System,” J. Appl. Phys.  41, 1898–1902, 2002.
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  11. M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik  112, 399–406 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, 1989).
  15. See: http://www.optics.arizona.edu/milster/optisca/DOCUMENTATION/uafdtd%20GUI%20MANUAL%20111906b.pdf.

2008 (1)

2007 (1)

2006 (1)

T. Chen, T. D. Milster, and S.-H. Yang, “Experimental investigation of photomask with near-field polarization imaging,” Proc. SPIE  6349, 634953 (2006).
[CrossRef]

2004 (1)

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE  5567, 887–893(2004).
[CrossRef]

2002 (2)

K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, S. Masuhara, M. Furuki, and M. Yamamoto, “Readout Method for Read Only Memory Signal and Air Gap Control Signal in a Near Field Optical Disc System,” J. Appl. Phys.  41, 1898–1902, 2002.
[CrossRef]

J. S. Jo, T. D. Milster, and J. K. Erwin, “Phase and amplitude apodization induced by focusing through an evanescent gap in a solid immersion lens microscope,” Opt. Eng.  41, 1866–1875 (2002).
[CrossRef]

2001 (1)

M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik  112, 399–406 (2001).
[CrossRef]

1998 (1)

1996 (1)

1992 (1)

G. S. Kino and S. M. Mansfield, “Solid immersion lens photon tunneling microscope,” (invited paper), Proc. SPIE  1556, 2–10(1992).
[CrossRef]

1989 (1)

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, 1989).

1986 (1)

1957 (1)

E. Wolf, “Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image Field,” Proc. R. Soc. London Ser. A  253, 349–357 (1957).
[CrossRef]

1953 (1)

H. H. Hopkins, “On the Diffraction Theory of Optical Images,” Proc. R. Soc. London Ser. A  217, 408–432 (1953).
[CrossRef]

Aspnes, E.

Brunner, R.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE  5567, 887–893(2004).
[CrossRef]

Buchda, G.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE  5567, 887–893(2004).
[CrossRef]

Chen, T.

T. Chen, T. D. Milster, S.-H. Yang, and D. Hansen, “Evanescent imaging with induced polarization by using a solid immersion lens,” Opt. Lett.  32, 124–126, January 2007.
[CrossRef]

T. Chen, T. D. Milster, and S.-H. Yang, “Experimental investigation of photomask with near-field polarization imaging,” Proc. SPIE  6349, 634953 (2006).
[CrossRef]

Erwin, J. K.

J. S. Jo, T. D. Milster, and J. K. Erwin, “Phase and amplitude apodization induced by focusing through an evanescent gap in a solid immersion lens microscope,” Opt. Eng.  41, 1866–1875 (2002).
[CrossRef]

Flagello, D. G.

Furuki, M.

K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, S. Masuhara, M. Furuki, and M. Yamamoto, “Readout Method for Read Only Memory Signal and Air Gap Control Signal in a Near Field Optical Disc System,” J. Appl. Phys.  41, 1898–1902, 2002.
[CrossRef]

Hansen, D.

Hopkins, H. H.

H. H. Hopkins, “On the Diffraction Theory of Optical Images,” Proc. R. Soc. London Ser. A  217, 408–432 (1953).
[CrossRef]

Horn, U.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE  5567, 887–893(2004).
[CrossRef]

Ishimoto, T.

K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, S. Masuhara, M. Furuki, and M. Yamamoto, “Readout Method for Read Only Memory Signal and Air Gap Control Signal in a Near Field Optical Disc System,” J. Appl. Phys.  41, 1898–1902, 2002.
[CrossRef]

Jo, J. S.

J. S. Jo, T. D. Milster, and J. K. Erwin, “Phase and amplitude apodization induced by focusing through an evanescent gap in a solid immersion lens microscope,” Opt. Eng.  41, 1866–1875 (2002).
[CrossRef]

Kino, G. S.

G. S. Kino and S. M. Mansfield, “Solid immersion lens photon tunneling microscope,” (invited paper), Proc. SPIE  1556, 2–10(1992).
[CrossRef]

Kondo, T.

K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, S. Masuhara, M. Furuki, and M. Yamamoto, “Readout Method for Read Only Memory Signal and Air Gap Control Signal in a Near Field Optical Disc System,” J. Appl. Phys.  41, 1898–1902, 2002.
[CrossRef]

Lang, M.

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, 1989).

Mansfield, S. M.

G. S. Kino and S. M. Mansfield, “Solid immersion lens photon tunneling microscope,” (invited paper), Proc. SPIE  1556, 2–10(1992).
[CrossRef]

Mansuripur, M.

Masuhara, S.

K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, S. Masuhara, M. Furuki, and M. Yamamoto, “Readout Method for Read Only Memory Signal and Air Gap Control Signal in a Near Field Optical Disc System,” J. Appl. Phys.  41, 1898–1902, 2002.
[CrossRef]

Menck, A.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE  5567, 887–893(2004).
[CrossRef]

Milster, T. D.

Nakaoki, A.

K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, S. Masuhara, M. Furuki, and M. Yamamoto, “Readout Method for Read Only Memory Signal and Air Gap Control Signal in a Near Field Optical Disc System,” J. Appl. Phys.  41, 1898–1902, 2002.
[CrossRef]

Rosenbluth, A. E.

Saito, K.

K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, S. Masuhara, M. Furuki, and M. Yamamoto, “Readout Method for Read Only Memory Signal and Air Gap Control Signal in a Near Field Optical Disc System,” J. Appl. Phys.  41, 1898–1902, 2002.
[CrossRef]

Steiner, R.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE  5567, 887–893(2004).
[CrossRef]

Totzeck, M.

M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik  112, 399–406 (2001).
[CrossRef]

Weissenberg, S.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE  5567, 887–893(2004).
[CrossRef]

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 (1957).
[CrossRef]

Yamamoto, M.

K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, S. Masuhara, M. Furuki, and M. Yamamoto, “Readout Method for Read Only Memory Signal and Air Gap Control Signal in a Near Field Optical Disc System,” J. Appl. Phys.  41, 1898–1902, 2002.
[CrossRef]

Yang, S.-H.

T. Chen, T. D. Milster, S.-H. Yang, and D. Hansen, “Evanescent imaging with induced polarization by using a solid immersion lens,” Opt. Lett.  32, 124–126, January 2007.
[CrossRef]

T. Chen, T. D. Milster, and S.-H. Yang, “Experimental investigation of photomask with near-field polarization imaging,” Proc. SPIE  6349, 634953 (2006).
[CrossRef]

Zibold, A. M.

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE  5567, 887–893(2004).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (1)

K. Saito, T. Ishimoto, T. Kondo, A. Nakaoki, S. Masuhara, M. Furuki, and M. Yamamoto, “Readout Method for Read Only Memory Signal and Air Gap Control Signal in a Near Field Optical Disc System,” J. Appl. Phys.  41, 1898–1902, 2002.
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

J. S. Jo, T. D. Milster, and J. K. Erwin, “Phase and amplitude apodization induced by focusing through an evanescent gap in a solid immersion lens microscope,” Opt. Eng.  41, 1866–1875 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optik (1)

M. Totzeck, “Numerical simulation of high-NA quantitative polarization microscopy and corresponding near-fields,” Optik  112, 399–406 (2001).
[CrossRef]

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 (1957).
[CrossRef]

H. H. Hopkins, “On the Diffraction Theory of Optical Images,” Proc. R. Soc. London Ser. A  217, 408–432 (1953).
[CrossRef]

Proc. SPIE (3)

G. S. Kino and S. M. Mansfield, “Solid immersion lens photon tunneling microscope,” (invited paper), Proc. SPIE  1556, 2–10(1992).
[CrossRef]

T. Chen, T. D. Milster, and S.-H. Yang, “Experimental investigation of photomask with near-field polarization imaging,” Proc. SPIE  6349, 634953 (2006).
[CrossRef]

R. Brunner, A. Menck, R. Steiner, G. Buchda, S. Weissenberg, U. Horn, and A. M. Zibold, “Immersion mask inspection with hybrid-microscopic systems at 193 nm,” Proc. SPIE  5567, 887–893(2004).
[CrossRef]

Other (2)

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, 1989).

See: http://www.optics.arizona.edu/milster/optisca/DOCUMENTATION/uafdtd%20GUI%20MANUAL%20111906b.pdf.

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Figures (11)

Fig. 1
Fig. 1

SIL imaging system.

Fig. 2
Fig. 2

Babinet’s principle is used to decompose the fields reflected from the sample into components that consist of reflection from different areas.

Fig. 3
Fig. 3

Abbe theory adapted to the high-NA microscope. An obliquely incident plane wave from a point light source is diffracted by the object according to the object structure. The diffracted orders are filtered by the stop and form the image by interference in the image plane.

Fig. 4
Fig. 4

Angle definitions for simulation. θ is the angle of incidence with respect to z axis. φ is the rotational angle about the z axis. ψ inc is the polarization angle ( ψ inc = 90 ° for s-polarization and ψ inc = 0 ° for p-polarization).

Fig. 5
Fig. 5

Reflectance angular distribution for native and induced polarizations. The white circle represents the critical angle. ( NA = 1.5 ; refractive indices of incidence medium and substrate are 1.85 and 1.0 at λ=550 nm, respectively; | A ill ( k ̂ ) | = 1 for all k ̂ ).

Fig. 6
Fig. 6

S A ( Δ χ ) , S A B j ( Δ χ ) , and S B j l ( Δ χ ) for a binary chrome grating on a quartz substrate ( n CR = 2.314 + i 3.136 , n QZ = 1.546 ) with grating height 1 nm and 100 nm at 550 nm for σ c = 0.1 , 0.5, and 1. J 12 scaled to the image plane is shown for comparison.

Fig. 7
Fig. 7

Normalized irradiance distributions at the image plane. (Chrome grating on a quartz substrate, n CR = 2.314 + i 3.136 , duty ratio is 1:1, and substrate is quartz, n QZ = 1.546 at 550 nm . Λ = 1.5 μ m . The light source is polarized with ψ SRC = 90 ° , σ c = 1 ).

Fig. 8
Fig. 8

Normalized magnitudes of m = f i r s t , zeroth, and + f i r s t orders in image plane as a function of incidence angle ( φ = 45 ° ) .

Fig. 9
Fig. 9

Normalized irradiance distributions at the image plane. (Chrome grating on a quartz substrate, n CR = 2.314 + i 3.136 , duty ratio is 1:1, and substrate is quartz, n QZ = 1.546 at 550 nm . Grating height = 100 nm . The light source is polarized with ψ SRC = 90 ° , σ c = 1 , NA = 1.48 ).

Fig. 10
Fig. 10

Normalized irradiance distributions at the image plane. (Quartz grating on a quartz substrate, duty ratio is 1:1, and substrate is quartz, n QZ = 1.546 at 550 nm . Grating height = 100 nm . The light source is polarized with ψ SRC = 90 ° , σ c = 1 , NA = 1.48 ).

Fig. 11
Fig. 11

Normalized irradiance distributions of native polarization at the image plane. (Quartz grating on a quartz substrate and chrome grating on a quartz substrate, duty ratio is 1:1, and substrate is quartz, n QZ = 1.546 and n CR = 2.314 + i 3.136 at 550 nm . Grating height is 100 nm . The light source is polarized with ψ SRC = 90 ° , σ c = 0.5 , NA = 1.48 ).

Equations (33)

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U R ( r ) = [ R 0 b 0 ( r ) + R 1 b 1 ( r ) + R 2 b 2 ( r ) ] U ill ( r ) ,
U R ( r ) = [ R 0 + ( R 1 R 0 ) b 1 ( r ) + ( R 2 R 0 ) b 2 ( r ) ] U ill ( r ) .
U ill ( r ) = A ill ( k ̂ ) exp ( i k r ) ,
A ill ( k ̂ ) = [ A s A p ] = 1 α 2 + β 2 [ β α α β ] A SRC ( k ̂ , ψ SRC ) ,
A SRC ( k ̂ , ψ SRC ) = A SRC ( k ̂ ) [ cos ψ SRC sin ψ SRC ] ,
U R ( r , k ̂ ) = R ( k ̂ ) U ill ( r ) = A R ( k ̂ ) exp ( i k r ) ,
A R ( k ̂ ) = R ( k ̂ ) A ill ( k ̂ ) ,
R ( k ̂ ) = [ R s ( k ̂ ) 0 0 R p ( k ̂ ) ] ,
A stop ( k ̂ ) = 1 α 2 + β 2 [ β α α β ] A R ( k ̂ ) .
A stop ( k ̂ ) = A stop native ( k ̂ ) x ̂ + A stop induced ( k ̂ ) y ̂ .
A stop native ( k ̂ ) = R native ( k ̂ ) A SRC ( k ̂ ) ,
R native ( k ̂ ) = 1 α 2 + β 2 [ β 2 R s ( k ̂ ) + α 2 R p ( k ̂ ) ] .
A stop induced ( k ̂ ) = R induced ( k ̂ ) A SRC ( k ̂ ) ,
R induced ( k ̂ ) = α β α 2 + β 2 [ R s ( k ̂ ) R p ( k ̂ ) ] .
U R ( r , k ̂ ) = { R 0 ( k ̂ ) j = 1 N [ R 0 ( k ̂ ) R j ( k ̂ ) ] b j ( r ) } A ill ( k ̂ ) exp ( i k r ) = { R 0 ( k ̂ ) j = 1 N [ R 0 ( k ̂ ) R j ( k ̂ ) ] b j ( r ) } [ β α α β ] A SRC ( k ̂ , ψ ill ) α 2 + β 2 exp ( i k r ) ,
U image ( r , k ̂ ) = n SIL n image f 2 f 1 1 α 2 + β 2 [ β α α β ] U R ( r m T , k ̂ ) h ( r ) ,
U image ( r , k ̂ ) = U image native ( r , k ̂ ) x ̂ + U image induced ( r , k ̂ ) y ̂ ,
U image native ( r , k ̂ ) = n SIL n image f 2 f 1 R N native ( r m T , k ̂ ) A SRC ( k ̂ ) exp ( i k r m T ) h ( r ) ,
U image induced ( r , k ̂ ) = n SIL n image f 2 f 1 R N induced ( r m T , k ̂ ) A SRC ( k ̂ ) exp ( i k r m T ) h ( r ) .
R N native ( r m T , k ̂ ) = 1 α 2 + β 2 ( [ β 2 R s , 0 ( k ̂ ) + α 2 R p , 0 ( k ̂ ) ] j = 1 N { β 2 [ R s , 0 ( k ̂ ) R s , j ( k ̂ ) ] + α 2 [ R p , 0 ( k ̂ ) R p , j ( k ̂ ) ] } b j ( r m T ) ) ,
R N induced ( r m T , k ̂ ) = α β α 2 + β 2 { [ R s , 0 ( k ̂ ) R p , 0 ( k ̂ ) ] j = 1 N [ R s , 0 ( k ̂ ) R p , 0 ( k ̂ ) + R s , j ( k ̂ ) R p , j ( k ̂ ) ] b j ( r m T ) } .
I image native ( r , k ̂ ) = C I n SIL n image ( f 2 f 1 ) 2 | R N native ( r m T , k ̂ ) A SRC ( k ̂ ) exp ( i k r m T ) h ( r ) | 2 ,
R A native ( r m T , k ̂ ) = 1 α 2 + β 2 [ β 2 R s , 0 ( k ̂ ) + α 2 R p , 0 ( k ̂ ) ] ,
R B , j native ( r m T , k ̂ ) = β 2 α 2 + β 2 [ R s , 0 ( k ̂ ) R s , j ( k ̂ ) ] + α 2 α 2 + β 2 [ R p , 0 ( k ̂ ) R p , j ( k ̂ ) ] ,
R A induced ( r m T , k ̂ ) = α β α 2 + β 2 [ R s , 0 ( k ̂ ) R p , 0 ( k ̂ ) ] ,
R B , j induced ( r m T , k ̂ ) = α β α 2 + β 2 [ R s , 0 ( k ̂ ) R p , 0 ( k ̂ ) + R s , j ( k ̂ ) R p , j ( k ̂ ) ] .
| R N native ( r image m T , k ̂ ) A SRC ( k ̂ ) exp ( i k r image m T ) h ( r image ) | 2 = | [ R A native ( k ̂ ) j = 1 N R B , j native ( k ̂ ) b j ( χ m T ) ] A SRC ( k ̂ ) exp ( i k χ m T ) h ( r image χ ) d χ | 2 = | R A native ( k ̂ ) | 2 | A SRC ( k ̂ ) | 2 exp [ i k ( χ χ ) m T ] h ( r image χ ) h * ( r image χ ) d χ d χ j = 1 N R A native ( k ̂ ) R B , j native * ( k ̂ ) b j ( χ m T ) | A SRC ( k ̂ ) | 2 × exp [ i k ( χ χ ) m T ] h ( r image χ ) h * ( r image χ ) d χ d χ j = 1 N R A native * ( k ̂ ) R B , j native ( k ̂ ) b j ( χ m T ) | A SRC ( k ̂ ) | 2 × exp [ i k ( χ χ ) m T ] h ( r image χ ) h * ( r image χ ) d χ d χ + j = 1 N l = 1 N R B , j native ( k ̂ ) R B , l native * ( k ̂ ) b j ( χ m T ) b l ( χ m T ) | A SRC ( k ̂ ) | 2 × exp [ i k ( χ χ ) m T ] h ( r image χ ) h * ( r image χ ) d χ d χ ,
I image native ( r image ) = I image native ( r image , k ̂ ) d k ̂ = C I n SIL n image ( f 2 f 1 ) 2 | R N native ( r m T , k ̂ ) A SRC ( k ̂ ) exp ( i k r m T ) h ( r ) | 2 d k ̂ .
| R N native ( r image m T , k ̂ ) A SRC ( k ̂ ) exp ( i k r image m T ) h ( r image ) | 2 d k ̂ = S A ( Δ χ ) h ( r image χ ) h * ( r image χ ) d χ d χ j = 1 N S A B j ( Δ χ ) b j ( χ m T ) h ( r image χ ) h * ( r image χ ) d χ d χ j = 1 N S A B j * ( Δ χ ) b j ( χ m T ) h ( r image χ ) h * ( r image χ ) d χ d χ + j = 1 N l = 1 N S B j l ( Δ χ ) b j ( χ m T ) b l ( χ m T ) h ( r image χ ) h * ( r image χ ) d χ d χ .
S A ( Δ χ ) = | R A native ( k ̂ ) | 2 | A SRC ( k ̂ ) | 2 exp ( i k Δ χ m T ) d k ̂ = F Δ χ m T λ 1 [ | R A native ( k ̂ ) | 2 | A SRC ( k ̂ ) | 2 ] ,
S A B j ( Δ χ ) = R A native ( k ̂ ) R B j native * ( k ̂ ) | A SRC ( k ̂ ) | 2 exp ( i k Δ χ m T ) d k ̂ = F Δ χ m T λ 1 [ R A native ( k ̂ ) R B j native * ( k ̂ ) | A SRC ( k ̂ ) | 2 ] ,
S B j l ( Δ χ ) = R B j native ( k ̂ ) R B l native * ( k ̂ ) | A SRC ( k ̂ ) | 2 exp ( i k Δ χ m T ) d k ̂ = F Δ χ m T λ 1 [ R B j native ( k ̂ ) R B l native * ( k ̂ ) | A SRC ( k ̂ ) | 2 ] ,
J 12 ( Δ χ m T ) = | A SRC ( k ̂ ) | 2 exp ( i k Δ χ m T ) d k ̂ .

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