Abstract

We report a theoretical study on the optical resolution of coherent nonlinear microscopy by means of double-sided Feynman diagrams. Through the use of the diagrams, we offer a simple technique to calculate the coherent transfer function (CTF), which is employed as the indicator of the optical resolution. In particular, we deal with the CTFs of coherent anti-Stokes Raman scattering (CARS) microscopy and stimulated Raman scattering (SRS) microscopy. Our results show that CARS and SRS microscopy possess nearly identical optical resolutions if a molecular-vibrational frequency of interest is assumed to be negligible compared with excitation photon energy. The peculiar image-formation properties of third-harmonic generation (THG) microscopy also can be explained by our technique.

© 2013 Optical Society of America

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References

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    [CrossRef]
  2. P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, “Second-harmonic imaging microscopy of living cells,” J. Biomed. Opt. 6, 277–286 (2001).
    [CrossRef]
  3. J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. 196, 325–330 (2001).
    [CrossRef]
  4. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third-harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
    [CrossRef]
  5. M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191, 266–274 (1998).
    [CrossRef]
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    [CrossRef]
  7. D. Yelin and Y. Silberberg, “Laser scanning third-harmonic generation microscopy in biology,” Opt. Express 5, 169–175 (1999).
    [CrossRef]
  8. M. D. Duncan, J. Reintjes, and T. J. Manuccia, “Scanning coherent anti-Stokes Raman microscope,” Opt. Lett. 7, 350–352 (1982).
    [CrossRef]
  9. A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational microscopy using coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
    [CrossRef]
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    [CrossRef]
  11. W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
    [CrossRef]
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    [CrossRef]
  13. G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science 330, 1368–1370 (2010).
    [CrossRef]
  14. M. Hashimoto and T. Araki, “Three-dimensional transfer functions of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. A 18, 771–776 (2001).
    [CrossRef]
  15. N. Fukutake, “Resolution properties of nonlinear optical microscopy,” J. Opt. Soc. Am. A 27, 1701–1707 (2010).
    [CrossRef]
  16. N. Fukutake, “Coherent transfer function of Fourier transform spectral interferometric coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. A 28, 1689–1694 (2011).
    [CrossRef]
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  18. R. W. Boyd, Nonlinear Optics (Academic, 1992).
  19. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1974).
  20. J. R. Sheppard and M. Gu, “The three-dimensional (3-D) transmission cross-coefficient for transmission imaging,” Optik 100, 155–158 (1995).
  21. J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002).
    [CrossRef]

2012 (1)

2011 (1)

2010 (2)

N. Fukutake, “Resolution properties of nonlinear optical microscopy,” J. Opt. Soc. Am. A 27, 1701–1707 (2010).
[CrossRef]

G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science 330, 1368–1370 (2010).
[CrossRef]

2009 (1)

2008 (1)

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

2002 (1)

2001 (3)

M. Hashimoto and T. Araki, “Three-dimensional transfer functions of coherent anti-Stokes Raman scattering microscopy,” J. Opt. Soc. Am. A 18, 771–776 (2001).
[CrossRef]

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, “Second-harmonic imaging microscopy of living cells,” J. Biomed. Opt. 6, 277–286 (2001).
[CrossRef]

J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. 196, 325–330 (2001).
[CrossRef]

1999 (3)

1998 (1)

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191, 266–274 (1998).
[CrossRef]

1997 (1)

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third-harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

1995 (1)

J. R. Sheppard and M. Gu, “The three-dimensional (3-D) transmission cross-coefficient for transmission imaging,” Optik 100, 155–158 (1995).

1986 (1)

1982 (1)

Araki, T.

Barad, Y.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third-harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1974).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Brakenhoff, G. J.

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191, 266–274 (1998).
[CrossRef]

Campagnola, P. J.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, “Second-harmonic imaging microscopy of living cells,” J. Biomed. Opt. 6, 277–286 (2001).
[CrossRef]

Cheng, J. X.

Clark, H. A.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, “Second-harmonic imaging microscopy of living cells,” J. Biomed. Opt. 6, 277–286 (2001).
[CrossRef]

Couderc, V.

Dake, F.

Deutsch, M.

Duncan, M. D.

Eisenberg, H.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third-harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

Freudiger, C. W.

G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science 330, 1368–1370 (2010).
[CrossRef]

Freudiger, W.

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

Freund, I.

Fukui, K.

Fukutake, N.

Gu, M.

J. R. Sheppard and M. Gu, “The three-dimensional (3-D) transmission cross-coefficient for transmission imaging,” Optik 100, 155–158 (1995).

Hamaguchi, H.

Hashimoto, M.

He, C.

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

Holtom, G. R.

G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science 330, 1368–1370 (2010).
[CrossRef]

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational microscopy using coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Horowitz, M.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third-harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

Itoh, K.

Kajiyama, S.

Kang, J. X.

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

Kano, H.

Leproux, P.

Lewis, A.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, “Second-harmonic imaging microscopy of living cells,” J. Biomed. Opt. 6, 277–286 (2001).
[CrossRef]

Loew, L. M.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, “Second-harmonic imaging microscopy of living cells,” J. Biomed. Opt. 6, 277–286 (2001).
[CrossRef]

Lu, S.

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

Manuccia, T. J.

Mertz, J.

J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. 196, 325–330 (2001).
[CrossRef]

Min, W.

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

Mohler, W. A.

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, “Second-harmonic imaging microscopy of living cells,” J. Biomed. Opt. 6, 277–286 (2001).
[CrossRef]

Moreaux, L.

J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. 196, 325–330 (2001).
[CrossRef]

Mukamel, S.

S. Mukamel, Principle of Nonlinear Optical Spectroscopy (Oxford University, 1995).

Muller, M.

J. A. Squier and M. Muller, “Third-harmonic generation imaging of laser-induced breakdown in glass,” Appl. Opt. 38, 5789–5794 (1999).
[CrossRef]

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191, 266–274 (1998).
[CrossRef]

Okuno, M.

Ozeki, Y.

Reichman, J.

G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science 330, 1368–1370 (2010).
[CrossRef]

Reintjes, J.

Saar, B. G.

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

Saar, G.

G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science 330, 1368–1370 (2010).
[CrossRef]

Segawa, H.

Sheppard, J. R.

J. R. Sheppard and M. Gu, “The three-dimensional (3-D) transmission cross-coefficient for transmission imaging,” Optik 100, 155–158 (1995).

Silberberg, Y.

D. Yelin and Y. Silberberg, “Laser scanning third-harmonic generation microscopy in biology,” Opt. Express 5, 169–175 (1999).
[CrossRef]

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third-harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

Squier, J.

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191, 266–274 (1998).
[CrossRef]

Squier, J. A.

Stanley, C. M.

G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science 330, 1368–1370 (2010).
[CrossRef]

Tsai, J. C.

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

Wilson, K. R.

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191, 266–274 (1998).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1974).

Xie, X. S.

G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science 330, 1368–1370 (2010).
[CrossRef]

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B 19, 1604–1610 (2002).
[CrossRef]

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational microscopy using coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Yelin, D.

Zumbusch, A.

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational microscopy using coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third-harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997).
[CrossRef]

J. Biomed. Opt. (1)

P. J. Campagnola, H. A. Clark, W. A. Mohler, A. Lewis, and L. M. Loew, “Second-harmonic imaging microscopy of living cells,” J. Biomed. Opt. 6, 277–286 (2001).
[CrossRef]

J. Microsc. (1)

M. Muller, J. Squier, K. R. Wilson, and G. J. Brakenhoff, “3D microscopy of transparent objects using third-harmonic generation,” J. Microsc. 191, 266–274 (1998).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

Opt. Commun. (1)

J. Mertz and L. Moreaux, “Second-harmonic generation by focused excitation of inhomogeneously distributed scatterers,” Opt. Commun. 196, 325–330 (2001).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Optik (1)

J. R. Sheppard and M. Gu, “The three-dimensional (3-D) transmission cross-coefficient for transmission imaging,” Optik 100, 155–158 (1995).

Phys. Rev. Lett. (1)

A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational microscopy using coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142–4145 (1999).
[CrossRef]

Science (2)

G. Saar, C. W. Freudiger, J. Reichman, C. M. Stanley, G. R. Holtom, and X. S. Xie, “Video-rate molecular imaging in vivo with stimulated Raman scattering,” Science 330, 1368–1370 (2010).
[CrossRef]

W. Freudiger, W. Min, B. G. Saar, S. Lu, G. R. Holtom, C. He, J. C. Tsai, J. X. Kang, and X. S. Xie, “Label-free biomedical imaging with high sensitivity by stimulated Raman scattering microscopy,” Science 322, 1857–1861 (2008).
[CrossRef]

Other (3)

S. Mukamel, Principle of Nonlinear Optical Spectroscopy (Oxford University, 1995).

R. W. Boyd, Nonlinear Optics (Academic, 1992).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1974).

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Figures (8)

Fig. 1.
Fig. 1.

Example of the double-sided Feynman diagram. ωi represents light frequencies. a, b, c, d, and e denote energy levels of a molecule of interest. In the energy level diagram, solid lines correspond to real levels and dotted lines correspond to virtual levels.

Fig. 2.
Fig. 2.

(a) Double-sided Feynman diagrams for CARS, SRL, and SRG signals. (b) Double-sided Feynman diagrams describing nonresonant backgrounds accompanied by CARS, SRL, and SRG signals.

Fig. 3.
Fig. 3.

Schematic of coherent nonlinear microscopy.

Fig. 4.
Fig. 4.

Cross section of the CTF for CARS.

Fig. 5.
Fig. 5.

Cross section of the CTF for SRS.

Fig. 6.
Fig. 6.

Double-sided Feynman diagram for THG.

Fig. 7.
Fig. 7.

Cross section of the CTF for THG.

Fig. 8.
Fig. 8.

xz cross-sectional images, obtained through confocal THG microscopy, of the spherical samples with (a) larger and (b) smaller diameters than the pump wavelength.

Equations (30)

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ItypeI(x)=|χ(i)(x+x)uex(x)ucol(xax)d3x|2a(xa)d3xa,
ItypeI(x)exp[i2πf·x]d3x=T(f,ff)X(i)(f)X(i)*(ff)d3f,
T(f,ff)=A(f2f1)Ucol(f1)Ucol*(f2)×Uex(f1f)Uex*(f2f+f)d3f1d3f2,
Uex(f)=[(PlPm)(Pn*Po*)](f),
(g1g2)(f)=g1(f)g2(ff)d3f
(g1g2)(f)=g1(f)g2(ff)d3f,
ItypeIC(x)=|χ(i)(x+x)uex(x)ucol(x)d3x|2,
TC(f,ff)=CTF(f)CTF*(ff)
CTF(f)=(UcolUex)(f)={Ucol[(PlPm)(Pn*Po*)]}(f),
Xcos(i)(f)=δ(f)+12{δ(ff0)+δ(f+f0)},
ItypeIcos(x)={T(0,0)+14T(f0,f0)+14T(f0,f0)}+12{T(f0,0)+T(0,f0)}exp(i2πf0·x)+12{T(0,f0)+T(f0,0)}exp(i2πf0·x)+14T(f0,f0)exp(i2π2f0·x)+14T(f0,f0)exp(i2π2f0·x),
TC(0,0)=|CTF(0)|2TC(f0,f0)=|CTF(f0)|2TC(f0,f0)=|CTF(f0)|2TC(f0,0)=CTF(f0)CTF*(0)TC(0,f0)=CTF*(f0)CTF(0)TC(f0,0)=CTF(f0)CTF*(0)TC(0,f0)=CTF*(f0)CTF(0)TC(f0,f0)=CTF(f0)CTF*(f0)TC(f0,f0)=CTF*(f0)CTF(f0).
TNC(f,ff)=|Ucol(f1)|2Uex(f1f)Uex*(f1f+f)d3f1.
TNC(0,0)=|Ucol(f1)|2|Uex(f1)|2d3f1TNC(f0,f0)=|Ucol(f1)|2|Uex(f1f0)|2d3f1TNC(f0,f0)=|Ucol(f1)|2|Uex(f1+f0)|2d3f1TNC(f0,0)=|Ucol(f1)|2Uex(f1f0)Uex*(f1)d3f1TNC(0,f0)=|Ucol(f1)|2Uex*(f1f0)Uex(f1)d3f1TNC(f0,0)=|Ucol(f1)|2Uex(f1+f0)Uex*(f1)d3f1TNC(0,f0)=|Ucol(f1)|2Uex*(f1+f0)Uex(f1)d3f1TNC(f0,f0)=|Ucol(f1)|2Uex(f1f0)Uex*(f1+f0)d3f1TNC(f0,f0)=|Ucol(f1)|2Uex*(f1f0)Uex(f1+f0)d3f1.
CTFCARS(f)={PCARS([PpuPpu]PS*)}(f),
ItypeII(x)=|αulo(xa)+χ(i)(x+x)uex(x)ucol(xax)d3x|2a(xa)d3xa,
ISRSNC(x)=|iulo(xa)+χSRS(3)(x+x)uex(x)ucol(xax)d3x|2d3xa{|ulo(xa)|2+iulo*(xa)χSRS(3)(x+x)uex(x)ucol(xax)d3xiulo(xa)χSRS(3)*(x+x)uex*(x)ucol*(xax)d3x}d3xa=c2|ucol(xa)|2d3xa+icχSRS(3)(x+x)uex(x)ucol(x)d3xicχSRS(3)*(x+x)uex*(x)ucol*(x)d3x,
icXSRS(3)(f)CTFSRS(f)icXSRS(3)*(f)CTFSRS*(f),
CTFSRL(f)={Ppu[(PpuPS)(PS*)]}(f)
CTFSRG(f)={PS[(PpuPS)(Ppu*)]}(f)
ISRSC(x)=|iulo(xa)+χSRS(3)(x+x)uex(x)ucol(xax)d3x|2δ(xa)d3xac2|ucol(0)|2+icucol*(0)χSRS(3)(x+x)uex(x)ucol(x)d3xicucol(0)χSRS(3)*(x+x)uex*(x)ucol*(x)d3x.
ICARSIINC(x)=|{χNRdc(3)+χNRac(3)(x+x)+χR(3)(x+x)}×uex(x)ucol(xax)d3x|2d3xa{χNRdc(3)}2|ucol(xa)|2d3xa+χNRdc(3){χNRac(3)(x+x)+χR(3)(x+x)}uex(x)ucol(x)d3x+χNRdc(3){χNRac(3)(x+x)+χR(3)(x+x)}*uex*(x){ucol(x)}*d3x,
FT{ICARSIINC}(f)={χNRdc(3)}2|ucol(xa)|2d3xaδ(f)+χNRdc(3){XNRac(3)(f)+XR(3)(f)}CTF(f)+χNRdc(3){XNRac(3)(f)+XR(3)(f)}*{CTF(f)}*
CTF(f)={PCARS[(PpuPpu)(PS*)]}(f),
ICARSIIC(x)=|{χNRdc(3)+χNRac(3)(x+x)+χR(3)(x+x)}×uex(x)ucol(xax)d3x|2δ(xa)d3xa{χNRdc(3)}2|C|2+χNRdc(3)C*{χNRac(3)(x+x)+χR(3)(x+x)}uex(x)ucol(x)d3x+χNRdc(3)C{χNRac(3)(x+x)+χR(3)(x+x)}*uex*(x)ucol*(x)d3x
FT{ICARSIIC}(f)={χNRdc(3)}2|C|2δ(f)+χNRdc(3)C*{XNRac(3)(f)+XR(3)(f)}CTFCARS(f)+χNRdc(3)C{XNRac(3)(f)+XR(3)(f)}*{CTFCARS(f)}*,
FT{ICARSIINC}(f){χNRdc(3)}2|ucol(xa)|2d3xaδ(f)+χNRdc(3){XNRac(3)(f)+XR(3)(f)}{CTF(f)+CTF*(f)}+iχNRdc(3)XR(3)(f){CTF(f)CTF*(f)},
cXSRS(3)(f){CTFSRS(f)+CTFSRS*(f)}+ic(XSRS(3)(f)+XNRSRS(3)(f)){CTFSRS(f)CTFSRS*(f)}.
AtypeI(x)exp[i2πf·x]d3x=CTF(f)X(3)(f),
CTF(f)={PTHG(PpuPpuPpu)}(f),

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