Abstract

We report a scheme to achieve resolution beyond the diffraction limit in spatial light interference microscopy (SLIM). By adding a grating to the optical path, the structured illumination technique can be used to improve the resolution by a factor of 2. We show that a direct application of the structured illumination technique, however, has proved to be unsuccessful. Through two crucial modifications, namely, one to the pupil plane of the objective and the other to the demodulation procedure, faithful phase information of the object is recovered and the resolution is improved by a factor of 2.

© 2012 Optical Society of America

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References

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  1. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
    [CrossRef]
  2. R. Liang, J. K. Erwin, and M. Mansuripur, “Variation on Zernike’s phase-contrast microscope,” Appl. Opt. 39, 2152–2158 (2000).
  3. C. J. Schwarz, Y. Kuznetsova, and S. Brueck, “Imaging interferometric microscopy,” Opt. Lett. 28, 1424–1426 (2003).
    [CrossRef]
  4. M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 1–50 (2010).
    [CrossRef]
  5. G. Popesu, L. P. Deflores, and J. C. Vaughan, “Fourier phase microscopy for investigation of biological structures and dynamics,” Opt. Lett. 29, 2503–2505 (2004).
    [CrossRef]
  6. T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30, 1165–1167 (2005).
    [CrossRef]
  7. G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31, 775–777 (2006).
    [CrossRef]
  8. Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19, 1016–1026 (2011).
    [CrossRef]
  9. H. Ding and G. Popescu, “Instantaneous spatial light interference microscopy,” Opt. Express 18, 1569–1575 (2010).
    [CrossRef]
  10. H. Ding, Z. Wang, X. Liang, S. A. Boppart, K. Tangella, and G. Popescu, “Measuring the scattering parameters of tissues from quantitative phase imaging of thin slices,” Opt. Lett. 36, 2281–2283 (2011).
    [CrossRef]
  11. Z. Wang, L. Millet, V. Chan, H. Ding, M. U. Gillette, R. Bashir, and G. Popescu, “Label-free intracellular transport measured by spatial light interference microscopy,” J. Biomed. Opt. 16, 026019 (2011).
  12. B. Huang, M. Bates, and X. Zhuang, “Super-resolution fluorescence microscopy,” Ann. Rev. Biochem. 78, 993–1016 (2009).
    [CrossRef]
  13. G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Ann. Rev. Phys. Chem. 61, 345–367 (2010).
    [CrossRef]
  14. S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
    [CrossRef]
  15. M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
    [CrossRef]
  16. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. A 217, 408–432 (1953).
  17. C. J. R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” J. Mod. Opt. 24, 1051–1073 (1977).
  18. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999), pp. 554–618.
  19. P. Mondal, “Phase contrast microscopy in partially coherent light,” Opt. Acta 15, 65–82 (1968).
    [CrossRef]
  20. L. Rayleigh, “On the accuracy of focus necessary for sensibly perfect definition,” in Scientific Papers, Vol. 1 (Cambridge University, 1899), pp. 430–432.
  21. J. Haldar, Z. Wang, G. Popescu, and Z.-P. Liang, “Label-free high-resolution imaging of live cells with deconvolved spatial light interference microscopy,” in Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE (2010), pp. 3382–3385.
  22. S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images” (Invited Paper), Proc. SPIE 7094, 709402 (2008).
  23. P. Hariharan, Basics of Interferometry, 2nd ed. (Academic, 2007), pp. 49–56.
  24. D. Brandner and G. Withers, http://cellimagelibrary.org/images/8735 .
  25. L. P. Yaroslavsky and H. J. Caulfield, “Deconvolution of multiple images of the same object,” Appl. Opt. 33, 2157–2162(1994).
    [CrossRef]
  26. S. D. Babacan, Z. Wang, M. Do, and G. Popescu, “Cell imaging beyond the diffraction limit using sparse deconvolution spatial light interference microscopy,” Biomed. Opt. Express 2, 1815–1827 (2011).
    [CrossRef]

2011 (4)

2010 (3)

H. Ding and G. Popescu, “Instantaneous spatial light interference microscopy,” Opt. Express 18, 1569–1575 (2010).
[CrossRef]

M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 1–50 (2010).
[CrossRef]

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Ann. Rev. Phys. Chem. 61, 345–367 (2010).
[CrossRef]

2009 (1)

B. Huang, M. Bates, and X. Zhuang, “Super-resolution fluorescence microscopy,” Ann. Rev. Biochem. 78, 993–1016 (2009).
[CrossRef]

2008 (1)

S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images” (Invited Paper), Proc. SPIE 7094, 709402 (2008).

2007 (1)

S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[CrossRef]

2006 (1)

2005 (1)

2004 (1)

2003 (1)

2000 (2)

R. Liang, J. K. Erwin, and M. Mansuripur, “Variation on Zernike’s phase-contrast microscope,” Appl. Opt. 39, 2152–2158 (2000).

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef]

1994 (1)

1977 (1)

C. J. R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” J. Mod. Opt. 24, 1051–1073 (1977).

1968 (1)

P. Mondal, “Phase contrast microscopy in partially coherent light,” Opt. Acta 15, 65–82 (1968).
[CrossRef]

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef]

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. A 217, 408–432 (1953).

Babacan, S. D.

Bashir, R.

Z. Wang, L. Millet, V. Chan, H. Ding, M. U. Gillette, R. Bashir, and G. Popescu, “Label-free intracellular transport measured by spatial light interference microscopy,” J. Biomed. Opt. 16, 026019 (2011).

Bates, M.

B. Huang, M. Bates, and X. Zhuang, “Super-resolution fluorescence microscopy,” Ann. Rev. Biochem. 78, 993–1016 (2009).
[CrossRef]

Boppart, S. A.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999), pp. 554–618.

Brueck, S.

Caulfield, H. J.

Chan, V.

Z. Wang, L. Millet, V. Chan, H. Ding, M. U. Gillette, R. Bashir, and G. Popescu, “Label-free intracellular transport measured by spatial light interference microscopy,” J. Biomed. Opt. 16, 026019 (2011).

Choudhury, A.

C. J. R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” J. Mod. Opt. 24, 1051–1073 (1977).

Dasari, R. R.

Davidson, M.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Ann. Rev. Phys. Chem. 61, 345–367 (2010).
[CrossRef]

Deflores, L. P.

Ding, H.

Do, M.

Erwin, J. K.

Feld, M. S.

Fienup, J. R.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images” (Invited Paper), Proc. SPIE 7094, 709402 (2008).

Gillette, M. U.

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19, 1016–1026 (2011).
[CrossRef]

Z. Wang, L. Millet, V. Chan, H. Ding, M. U. Gillette, R. Bashir, and G. Popescu, “Label-free intracellular transport measured by spatial light interference microscopy,” J. Biomed. Opt. 16, 026019 (2011).

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef]

Haldar, J.

J. Haldar, Z. Wang, G. Popescu, and Z.-P. Liang, “Label-free high-resolution imaging of live cells with deconvolved spatial light interference microscopy,” in Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE (2010), pp. 3382–3385.

Hariharan, P.

P. Hariharan, Basics of Interferometry, 2nd ed. (Academic, 2007), pp. 49–56.

Hell, S. W.

S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. A 217, 408–432 (1953).

Huang, B.

B. Huang, M. Bates, and X. Zhuang, “Super-resolution fluorescence microscopy,” Ann. Rev. Biochem. 78, 993–1016 (2009).
[CrossRef]

Ikeda, T.

Kim, M. K.

M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 1–50 (2010).
[CrossRef]

Kuznetsova, Y.

Liang, R.

Liang, X.

Liang, Z.-P.

J. Haldar, Z. Wang, G. Popescu, and Z.-P. Liang, “Label-free high-resolution imaging of live cells with deconvolved spatial light interference microscopy,” in Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE (2010), pp. 3382–3385.

Lippincott-Schwartz, J.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Ann. Rev. Phys. Chem. 61, 345–367 (2010).
[CrossRef]

Manley, S.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Ann. Rev. Phys. Chem. 61, 345–367 (2010).
[CrossRef]

Mansuripur, M.

Millet, L.

Z. Wang, L. Millet, V. Chan, H. Ding, M. U. Gillette, R. Bashir, and G. Popescu, “Label-free intracellular transport measured by spatial light interference microscopy,” J. Biomed. Opt. 16, 026019 (2011).

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19, 1016–1026 (2011).
[CrossRef]

Mir, M.

Mondal, P.

P. Mondal, “Phase contrast microscopy in partially coherent light,” Opt. Acta 15, 65–82 (1968).
[CrossRef]

Patterson, G.

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Ann. Rev. Phys. Chem. 61, 345–367 (2010).
[CrossRef]

Popescu, G.

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19, 1016–1026 (2011).
[CrossRef]

S. D. Babacan, Z. Wang, M. Do, and G. Popescu, “Cell imaging beyond the diffraction limit using sparse deconvolution spatial light interference microscopy,” Biomed. Opt. Express 2, 1815–1827 (2011).
[CrossRef]

Z. Wang, L. Millet, V. Chan, H. Ding, M. U. Gillette, R. Bashir, and G. Popescu, “Label-free intracellular transport measured by spatial light interference microscopy,” J. Biomed. Opt. 16, 026019 (2011).

H. Ding, Z. Wang, X. Liang, S. A. Boppart, K. Tangella, and G. Popescu, “Measuring the scattering parameters of tissues from quantitative phase imaging of thin slices,” Opt. Lett. 36, 2281–2283 (2011).
[CrossRef]

H. Ding and G. Popescu, “Instantaneous spatial light interference microscopy,” Opt. Express 18, 1569–1575 (2010).
[CrossRef]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31, 775–777 (2006).
[CrossRef]

T. Ikeda, G. Popescu, R. R. Dasari, and M. S. Feld, “Hilbert phase microscopy for investigating fast dynamics in transparent systems,” Opt. Lett. 30, 1165–1167 (2005).
[CrossRef]

J. Haldar, Z. Wang, G. Popescu, and Z.-P. Liang, “Label-free high-resolution imaging of live cells with deconvolved spatial light interference microscopy,” in Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE (2010), pp. 3382–3385.

Popesu, G.

Rayleigh, L.

L. Rayleigh, “On the accuracy of focus necessary for sensibly perfect definition,” in Scientific Papers, Vol. 1 (Cambridge University, 1899), pp. 430–432.

Rogers, J.

Schwarz, C. J.

Sheppard, C. J. R.

C. J. R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” J. Mod. Opt. 24, 1051–1073 (1977).

Shroff, S. A.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images” (Invited Paper), Proc. SPIE 7094, 709402 (2008).

Tangella, K.

Unarunotai, S.

Vaughan, J. C.

Wang, Z.

H. Ding, Z. Wang, X. Liang, S. A. Boppart, K. Tangella, and G. Popescu, “Measuring the scattering parameters of tissues from quantitative phase imaging of thin slices,” Opt. Lett. 36, 2281–2283 (2011).
[CrossRef]

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19, 1016–1026 (2011).
[CrossRef]

S. D. Babacan, Z. Wang, M. Do, and G. Popescu, “Cell imaging beyond the diffraction limit using sparse deconvolution spatial light interference microscopy,” Biomed. Opt. Express 2, 1815–1827 (2011).
[CrossRef]

Z. Wang, L. Millet, V. Chan, H. Ding, M. U. Gillette, R. Bashir, and G. Popescu, “Label-free intracellular transport measured by spatial light interference microscopy,” J. Biomed. Opt. 16, 026019 (2011).

J. Haldar, Z. Wang, G. Popescu, and Z.-P. Liang, “Label-free high-resolution imaging of live cells with deconvolved spatial light interference microscopy,” in Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE (2010), pp. 3382–3385.

Williams, D. R.

S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images” (Invited Paper), Proc. SPIE 7094, 709402 (2008).

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999), pp. 554–618.

Yaroslavsky, L. P.

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef]

Zhuang, X.

B. Huang, M. Bates, and X. Zhuang, “Super-resolution fluorescence microscopy,” Ann. Rev. Biochem. 78, 993–1016 (2009).
[CrossRef]

Ann. Rev. Biochem. (1)

B. Huang, M. Bates, and X. Zhuang, “Super-resolution fluorescence microscopy,” Ann. Rev. Biochem. 78, 993–1016 (2009).
[CrossRef]

Ann. Rev. Phys. Chem. (1)

G. Patterson, M. Davidson, S. Manley, and J. Lippincott-Schwartz, “Superresolution imaging using single-molecule localization,” Ann. Rev. Phys. Chem. 61, 345–367 (2010).
[CrossRef]

Appl. Opt. (2)

Biomed. Opt. Express (1)

J. Biomed. Opt. (1)

Z. Wang, L. Millet, V. Chan, H. Ding, M. U. Gillette, R. Bashir, and G. Popescu, “Label-free intracellular transport measured by spatial light interference microscopy,” J. Biomed. Opt. 16, 026019 (2011).

J. Microsc. (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[CrossRef]

J. Mod. Opt. (1)

C. J. R. Sheppard and A. Choudhury, “Image formation in the scanning microscope,” J. Mod. Opt. 24, 1051–1073 (1977).

Opt. Acta (1)

P. Mondal, “Phase contrast microscopy in partially coherent light,” Opt. Acta 15, 65–82 (1968).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Proc. R. Soc. A (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. A 217, 408–432 (1953).

Proc. SPIE (1)

S. A. Shroff, J. R. Fienup, and D. R. Williams, “OTF compensation in structured illumination superresolution images” (Invited Paper), Proc. SPIE 7094, 709402 (2008).

Science (2)

S. W. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[CrossRef]

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef]

SPIE Rev. (1)

M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 1–50 (2010).
[CrossRef]

Other (5)

P. Hariharan, Basics of Interferometry, 2nd ed. (Academic, 2007), pp. 49–56.

D. Brandner and G. Withers, http://cellimagelibrary.org/images/8735 .

L. Rayleigh, “On the accuracy of focus necessary for sensibly perfect definition,” in Scientific Papers, Vol. 1 (Cambridge University, 1899), pp. 430–432.

J. Haldar, Z. Wang, G. Popescu, and Z.-P. Liang, “Label-free high-resolution imaging of live cells with deconvolved spatial light interference microscopy,” in Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE (2010), pp. 3382–3385.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999), pp. 554–618.

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Figures (6)

Fig. 1.
Fig. 1.

A simplified sketch for the setup of a spatial light interference microscope. The source plane and the pupil plane are Fourier planes with spatial frequency coordinates (fx, fy). The sample plane and the sensor plane are image planes with spatial coordinates (x, y). In the pupil plane the gray area is a conjugate image of the source, and both amplitude attenuation and phase modulation exist within this area.

Fig. 2.
Fig. 2.

Imaging results by (a) SLIM, (b) SLIM+SI, direct (c) SLIM+SI, adapted. (d) Cross-sections of the image in the dashed boxes (corresponding to element 6 in Group 1) in (a)–(c) show that the contrast of the adapted SLIM+SI is better than the direct SI+SLIM.

Fig. 3.
Fig. 3.

Proposed setups for xSLIM where the grating is relayed to the sample plane through a 4F system. The dashed box indicates the add-on module for SLIM. The dash-dotted box indicates the add-on module for xSLIM. (a) Transmission mode, (b) reflection mode.

Fig. 4.
Fig. 4.

Images of randomly positioned bead pairs by (a) SLIM and (b) xSLIM. The two beads denoted by white arrows are 0.5μm apart. The cross-sections of the images of these two beads by SLIM, xSLIM, and the original sample are shown in (c).

Fig. 5.
Fig. 5.

Imaging results of xSLIM when the signal-to-noise ratio in the image plane is (a) 104, (b) 103, and (c) 102. The zoomed-in version of the two beads denoted by the white arrows in (a)–(c) are shown in (d)–(f), respectively. The scale bars in (a) and (d) represent 2.5μm and 0.25μm, respectively.

Fig. 6.
Fig. 6.

Imaging results of a neuron-cell by xSLIM where contrast of the grating and the signal-to-noise ratio are (a) m=0.01, SNR=1000; (b) m=0.01, SNR=100; (c) m=0.001, SNR=1000; (d) m=0.001, SNR=100, respectively. Scale bar in (a) represents 10μm.

Equations (35)

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Is(fs)={1|fs|fs+[ε,ε],0elsewhere;
I(r)=|T(r)ei2π(fs·r)p(r)|2dfs,
T=U1+U2,
I˜(f)=[T˜(f+fs)H][T˜(f+fs)H]dfs,
fc(SLIM)=fo+fs.
H(f;ϕ)=aeiϕH1(f)+H2(f);
I˜(f;ϕ)=a2I˜11+I˜22+aeiϕI˜12+aeiϕI˜12*,
I˜ij(f)=[T˜(f+fs)Hi(f)][T˜(f+fs)Hj(f)]dfs,i,j=1,2.
I(r;ϕ)=a2I11+I22+aeiϕI12+aeiϕI12*.
I(r;ϕ)=|aU1eiϕ+U2|2=|U1||aeiϕ+βeϕ12|2,
ϕ12=arctan(I(r;π/2)I(r;π/2)I(r;0)I(r;π)),
|I12|=(I(r;π/2)I(r;π/2))2+(I(r;0)I(r;π))24a.
I=a2I11+I22=I(r;π/2)+I(r;π/2)+I(r;0)+I(r;π)4.
β(r)=|U2U1|=II24a2|I12|22|I12|.
U(SLIM)(r)=I11(1+βeiϕ12),
ϕ(SLIM)(r)=arctan(imag(U(SLIM))real(U(SLIM))).
d(SLIM)=1.22λNAo+NAc=1.22fc(SLIM).
T(r)=T(r)(1+mcos(2πfg·r+θ)),
T˜(f)=T˜(f)+m2eiθT˜(f+fg)+m2eiθT˜(ffg).
I˜(f;θ)=[T˜(f+fs)H][T˜(f+fs)H]dfs,=I˜0(f)+m2eiθI˜+1+m2eiθI˜1+m24ei2θI˜+2+m24ei2θI˜2,
I˜0={i=11[T˜(f+ifg+fs)H][T˜(f+ifg+fs)H]}dfs,
I˜+1={[T˜(f+fg+fs)H][T˜(f+fs)H]+[T˜(f+fs)H][T˜(ffg+fs)H]}dfs,
I˜1={[T˜(ffg+fs)H][T˜(f+fs)H]+[T˜(f+fs)H][T˜(f+fg+fs)H]}dfs,
I˜+2=[T˜(f+fg+fs)H][T˜(ffg+fs)H]dfs,
I˜2=[T˜(ffg+fs)H][T˜(f+fg+fs)H]dfs.
I˜(f)=I˜0(f)+I˜+1(ffg)+I˜1(f+fg)+I˜+2(f2fg)+I˜2(f+2fg).
H=aeiϕH1+H2,
H1={1where|f+(0,±fg)|=|fs|,0elsewhere;
U(r;θ)=U(1+mcos(2πfg·r+θ)),
U˜(f;θ)=U˜(f)+m2eiθU˜(f+fg)+m2eiθU˜(ffg).
[U˜(f;θ1)U˜(f;θ2)U˜(f;θ3)]=[eiθ11eiθ1eiθ21eiθ2eiθ31eiθ3][m2U˜(ffg)U˜(f)m2U˜(f+fg)].
U˜(xSLIM)=U˜(f)+U˜(f)+U˜+(f).
max(fc(xSLIM))=2(fo+fs)=2fc(SLIM).
U(xSLIM)(r)=F1{U˜(xSLIM)}.
ϕ(xSLIM)(r)=arctan(imag(U(xSLIM))real(U(xSLIM))),

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