Abstract

We propose to use intensity correlation microscopy in combination with structured illumination to image quantum emitters that exhibit antibunching with a spatial resolution reaching far beyond the Rayleigh limit. Combining intensity measurements and intensity autocorrelations up to order m creates an effective PSF with an FWHM shrunk by the factor m. Structured illumination microscopy, on the other hand, introduces a resolution improvement of factor 2 by use of the principle of moiré fringes. Here, we show that for linear low-intensity excitation and linear optical detection, the simultaneous use of both techniques leads to a theoretically unlimited resolution power, with the improvement scaling favorably as m+m, dependent on the correlation order m. Hence, this technique should be of interest in microscopy for imaging a variety of samples, including biological ones. We present the underlying theory and simulations, demonstrating the highly increased spatial superresolution, and point out the requirements for an experimental implementation.

© 2017 Optical Society of America

Full Article  |  PDF Article
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  39. K. O’Holleran and M. Shaw, “Optimized approaches for optical sectioning and resolution enhancement in 2D structured illumination microscopy,” Biomed. Opt. Express 5, 2580–2590 (2014).
    [Crossref]
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    [Crossref]
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    [Crossref]
  42. B. E. Urban, J. Yi, S. Chen, B. Dong, Y. Zhu, S. H. DeVries, V. Backman, and H. F. Zhang, “Super-resolution two-photon microscopy via scanning patterned illumination,” Phys. Rev. E 91, 042703 (2015).
    [Crossref]

2016 (2)

A. Classen, F. Waldmann, S. Giebel, R. Schneider, D. Bhatti, T. Mehringer, and J. von Zanthier, “Superresolving imaging of arbitrary one-dimensional arrays of thermal light sources using multiphoton interference,” Phys. Rev. Lett. 117, 253601 (2016).
[Crossref]

N. Chakrova, B. Rieger, and S. Stallinga, “Deconvolution methods for structured illumination microscopy,” J. Opt. Soc. Am. A 33, B12–B20 (2016).
[Crossref]

2015 (3)

B. E. Urban, J. Yi, S. Chen, B. Dong, Y. Zhu, S. H. DeVries, V. Backman, and H. F. Zhang, “Super-resolution two-photon microscopy via scanning patterned illumination,” Phys. Rev. E 91, 042703 (2015).
[Crossref]

S. W. Hell, “Nanoscopy with focused light (Nobel lecture),” Angew. Chem. Int. Ed. 54, 8054–8066 (2015).
[Crossref]

J. A. Miles, D. Das, Z. J. Simmons, and D. D. Yavuz, “Localization of atomic excitation beyond the diffraction limit using electromagnetically induced transparency,” Phys. Rev. A 92, 033838 (2015).
[Crossref]

2014 (4)

J. H. Park, S.-W. Lee, E. S. Lee, and J. Y. Lee, “A method for super-resolved CARS microscopy with structured illumination in two dimensions,” Opt. Express 22, 9854–9870 (2014).
[Crossref]

X. Zeng, M. Al-Amri, and M. S. Zubairy, “Nanometer-scale microscopy via graphene plasmons,” Phys. Rev. B 90, 235418 (2014).
[Crossref]

D. Gatto Monticone, K. Katamadze, P. Traina, E. Moreva, J. Forneris, I. Ruo-Berchera, P. Olivero, I. P. Degiovanni, G. Brida, and M. Genovese, “Beating the Abbe diffraction limit in confocal microscopy via nonclassical photon statistics,” Phys. Rev. Lett. 113, 143602 (2014).
[Crossref]

K. O’Holleran and M. Shaw, “Optimized approaches for optical sectioning and resolution enhancement in 2D structured illumination microscopy,” Biomed. Opt. Express 5, 2580–2590 (2014).
[Crossref]

2013 (2)

O. Schwartz, J. M. Levitt, R. Tenne, S. Itzhakov, Z. Deutsch, and D. Oron, “Superresolution microscopy with quantum emitters,” Nano Lett. 13, 5832–5836 (2013).
[Crossref]

J. A. Miles, Z. J. Simmons, and D. D. Yavuz, “Subwavelength localization of atomic excitation using electromagnetically induced transparency,” Phys. Rev. X 3, 031014 (2013).
[Crossref]

2012 (4)

T. Dertinger, J. Xu, O. F. Naini, R. Vogel, and S. Weiss, “Sofi-based 3D superresolution sectioning with a widefield microscope,” Opt. Nanosc. 1, 2 (2012).
[Crossref]

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (BSOFI),” Opt. Nanosc. 1, 4 (2012).
[Crossref]

S. Oppel, T. Büttner, P. Kok, and J. von Zanthier, “Superresolving multiphoton interferences with independent light sources,” Phys. Rev. Lett. 109, 233603 (2012).
[Crossref]

O. Schwartz and D. Oron, “Improved resolution in fluorescence microscopy using quantum correlations,” Phys. Rev. A 85, 033812 (2012).
[Crossref]

2010 (3)

2009 (1)

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. USA 106, 22287–22292 (2009).
[Crossref]

2008 (2)

M. Kiffner, J. Evers, and M. S. Zubairy, “Resonant interferometric lithography beyond the diffraction limit,” Phys. Rev. Lett. 100, 073602 (2008).
[Crossref]

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref]

2007 (1)

C. Thiel, T. Bastin, J. Martin, E. Solano, J. von Zanthier, and G. S. Agarwal, “Quantum imaging with incoherent photons,” Phys. Rev. Lett. 99, 133603 (2007).
[Crossref]

2006 (4)

S. T. Hess, T. P. Girirajan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation,” Biophys. J. 91, 4258–4272 (2006).
[Crossref]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods 3, 793–796 (2006).
[Crossref]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B 39, 3437–3446 (2006).
[Crossref]

2005 (2)

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA 102, 13081–13086 (2005).
[Crossref]

M. Hofmann, C. Eggeling, S. Jakobs, and S. W. Hell, “Breaking the diffraction barrier in fluorescence microscopy at low light intensities by using reversibly photoswitchable proteins,” Proc. Natl. Acad. Sci. USA 102, 17565–17569 (2005).
[Crossref]

2000 (4)

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]

B. Lounis, H. Bechtel, D. Gerion, P. Alivisatos, and W. Moerner, “Photon antibunching in single CDSE/ZNS quantum dot fluorescence,” Chem. Phys. Lett. 329, 399–404 (2000).
[Crossref]

R. Brouri, A. Beveratos, J.-P. Poizat, and P. Grangier, “Photon antibunching in the fluorescence of individual color centers in diamond,” Opt. Lett. 25, 1294–1296 (2000).
[Crossref]

P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto, “Quantum correlation among photons from a single quantum dot at room temperature,” Nature 406, 968–970 (2000).
[Crossref]

1997 (1)

R. M. Dickson, A. B. Cubitt, R. Y. Tsien, and W. E. Moerner, “On/off blinking and switching behaviour of single molecules of green fluorescent protein,” Nature 388, 355–358 (1997).
[Crossref]

1995 (1)

S. W. Hell and M. Kroug, “Ground-state-depletion fluorescence microscopy: A concept for breaking the diffraction resolution limit,” Appl. Phys. B 60, 495–497 (1995).
[Crossref]

1994 (1)

1992 (1)

T. Basché, W. E. Moerner, M. Orrit, and H. Talon, “Photon antibunching in the fluorescence of a single dye molecule trapped in a solid,” Phys. Rev. Lett. 69, 1516–1519 (1992).
[Crossref]

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[Crossref]

1963 (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[Crossref]

1879 (1)

L. Rayleigh, “XXXI. Investigations in optics, with special reference to the spectroscope,” Philos. Mag. 8(49), 261–274 (1879).
[Crossref]

1873 (1)

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Mikrosk. Anat. 9, 413–418 (1873).
[Crossref]

Abbe, E.

E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Mikrosk. Anat. 9, 413–418 (1873).
[Crossref]

Agard, D. A.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref]

Agarwal, G. S.

K. T. Kapale and G. S. Agarwal, “Subnanoscale resolution for microscopy via coherent population trapping,” Opt. Lett. 35, 2792–2794 (2010).
[Crossref]

C. Thiel, T. Bastin, J. Martin, E. Solano, J. von Zanthier, and G. S. Agarwal, “Quantum imaging with incoherent photons,” Phys. Rev. Lett. 99, 133603 (2007).
[Crossref]

G. S. Agarwal and K. T. Kapale, “Subwavelength atom localization via coherent population trapping,” J. Phys. B 39, 3437–3446 (2006).
[Crossref]

G. S. Agarwal, Quantum Optics (Cambridge University, 2012).

Al-Amri, M.

X. Zeng, M. Al-Amri, and M. S. Zubairy, “Nanometer-scale microscopy via graphene plasmons,” Phys. Rev. B 90, 235418 (2014).
[Crossref]

Z. Liao, M. Al-Amri, and M. Suhail Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett. 105, 183601 (2010).
[Crossref]

Alivisatos, P.

B. Lounis, H. Bechtel, D. Gerion, P. Alivisatos, and W. Moerner, “Photon antibunching in single CDSE/ZNS quantum dot fluorescence,” Chem. Phys. Lett. 329, 399–404 (2000).
[Crossref]

Backman, V.

B. E. Urban, J. Yi, S. Chen, B. Dong, Y. Zhu, S. H. DeVries, V. Backman, and H. F. Zhang, “Super-resolution two-photon microscopy via scanning patterned illumination,” Phys. Rev. E 91, 042703 (2015).
[Crossref]

Basché, T.

T. Basché, W. E. Moerner, M. Orrit, and H. Talon, “Photon antibunching in the fluorescence of a single dye molecule trapped in a solid,” Phys. Rev. Lett. 69, 1516–1519 (1992).
[Crossref]

Bastin, T.

C. Thiel, T. Bastin, J. Martin, E. Solano, J. von Zanthier, and G. S. Agarwal, “Quantum imaging with incoherent photons,” Phys. Rev. Lett. 99, 133603 (2007).
[Crossref]

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (storm),” Nat. Methods 3, 793–796 (2006).
[Crossref]

Bechtel, H.

B. Lounis, H. Bechtel, D. Gerion, P. Alivisatos, and W. Moerner, “Photon antibunching in single CDSE/ZNS quantum dot fluorescence,” Chem. Phys. Lett. 329, 399–404 (2000).
[Crossref]

Berclaz, C.

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (BSOFI),” Opt. Nanosc. 1, 4 (2012).
[Crossref]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Beveratos, A.

Bhatti, D.

A. Classen, F. Waldmann, S. Giebel, R. Schneider, D. Bhatti, T. Mehringer, and J. von Zanthier, “Superresolving imaging of arbitrary one-dimensional arrays of thermal light sources using multiphoton interference,” Phys. Rev. Lett. 117, 253601 (2016).
[Crossref]

Bocchio, N. L.

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (BSOFI),” Opt. Nanosc. 1, 4 (2012).
[Crossref]

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Brida, G.

D. Gatto Monticone, K. Katamadze, P. Traina, E. Moreva, J. Forneris, I. Ruo-Berchera, P. Olivero, I. P. Degiovanni, G. Brida, and M. Genovese, “Beating the Abbe diffraction limit in confocal microscopy via nonclassical photon statistics,” Phys. Rev. Lett. 113, 143602 (2014).
[Crossref]

Brouri, R.

Buratto, S. K.

P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto, “Quantum correlation among photons from a single quantum dot at room temperature,” Nature 406, 968–970 (2000).
[Crossref]

Büttner, T.

S. Oppel, T. Büttner, P. Kok, and J. von Zanthier, “Superresolving multiphoton interferences with independent light sources,” Phys. Rev. Lett. 109, 233603 (2012).
[Crossref]

Cande, W. Z.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref]

Carlton, P. M.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref]

Carson, P. J.

P. Michler, A. Imamoglu, M. D. Mason, P. J. Carson, G. F. Strouse, and S. K. Buratto, “Quantum correlation among photons from a single quantum dot at room temperature,” Nature 406, 968–970 (2000).
[Crossref]

Chakrova, N.

Chen, S.

B. E. Urban, J. Yi, S. Chen, B. Dong, Y. Zhu, S. H. DeVries, V. Backman, and H. F. Zhang, “Super-resolution two-photon microscopy via scanning patterned illumination,” Phys. Rev. E 91, 042703 (2015).
[Crossref]

Classen, A.

A. Classen, F. Waldmann, S. Giebel, R. Schneider, D. Bhatti, T. Mehringer, and J. von Zanthier, “Superresolving imaging of arbitrary one-dimensional arrays of thermal light sources using multiphoton interference,” Phys. Rev. Lett. 117, 253601 (2016).
[Crossref]

Colyer, R.

T. Dertinger, R. Colyer, G. Iyer, S. Weiss, and J. Enderlein, “Fast, background-free, 3D super-resolution optical fluctuation imaging (SOFI),” Proc. Natl. Acad. Sci. USA 106, 22287–22292 (2009).
[Crossref]

Cubitt, A. B.

R. M. Dickson, A. B. Cubitt, R. Y. Tsien, and W. E. Moerner, “On/off blinking and switching behaviour of single molecules of green fluorescent protein,” Nature 388, 355–358 (1997).
[Crossref]

Das, D.

J. A. Miles, D. Das, Z. J. Simmons, and D. D. Yavuz, “Localization of atomic excitation beyond the diffraction limit using electromagnetically induced transparency,” Phys. Rev. A 92, 033838 (2015).
[Crossref]

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313, 1642–1645 (2006).
[Crossref]

Degiovanni, I. P.

D. Gatto Monticone, K. Katamadze, P. Traina, E. Moreva, J. Forneris, I. Ruo-Berchera, P. Olivero, I. P. Degiovanni, G. Brida, and M. Genovese, “Beating the Abbe diffraction limit in confocal microscopy via nonclassical photon statistics,” Phys. Rev. Lett. 113, 143602 (2014).
[Crossref]

Dellagiacoma, C.

S. Geissbuehler, N. L. Bocchio, C. Dellagiacoma, C. Berclaz, M. Leutenegger, and T. Lasser, “Mapping molecular statistics with balanced super-resolution optical fluctuation imaging (BSOFI),” Opt. Nanosc. 1, 4 (2012).
[Crossref]

Denk, W.

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
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Figures (6)

Fig. 1.
Fig. 1.

(a) Three equidistant emitters, separated by d = 0.7 d R , were chosen, where the black, blue, green, and red curves, respectively, represent the intensity G ( 1 ) ( r ) , the intensity squared [ G ( 1 ) ( r ) ] 2 , the CM 2 ( r ) signal of Eq. (3), and the scaled second-order correlation function 1 2 G ( 2 ) ( r ) . (b) The same setup with added structured illumination I str ( r ) = [ 1 2 + 1 2 cos ( k 0 r + φ ) ] for phase φ = 0 , such that the black and green curves now represent Eqs. (4) and (7). For the green curve, the effectively squared structured illumination I str ( r = 0.7 ) 2 = 0.65 addresses the outer emitters, which are then weaker than the central emitter. The latter one can therefore be better resolved. Varying the phase φ addresses different emitters sequentially. Note that the image plane coordinate r is given in units d R .

Fig. 2.
Fig. 2.

Schematic setup to combine SIM and CM to obtain the new SIQCM technique. The independent quantum emitters radiate fluorescent light (red) after being excited by a structured illumination standing-wave pattern (green). The SIQCM signal is obtained by post-processing a series of images that are captured by a CCD camera in the image plane. For details, see the text in Section 4.

Fig. 3.
Fig. 3.

(a)–(c) Comparison of the observable regions in reciprocal space provided by ordinary SIM, SIQCM 2 , and SIQCM 3 (left to right), where α = 3 , α = 4 , and α = 6 orientations of the structured illumination pattern were chosen, respectively. The axes are normalized by the modulus of the highest spatial frequency k max 1 transmitted through the imaging system. The green disks stem from the densities n ˜ ( k ± k 0 ) [cf. Eq. (5)], while the red and black disks originate from the higher harmonics arising in the SIQCM m signals [cf. Eq. (8)]. Note that the size of individual disks is enlarged from (a) to (c).

Fig. 4.
Fig. 4.

Comparison of the resulting final images utilizing ordinary intensity measurements G ( 1 ) ( r ) , CM 2 ( r ) , SIM, and SIQCM 2 ( r ) , imaging three 3 × 3 arrays with grid constants d = 1.0 d R , d = 0.5 d R , and d = 0.29 d R , and six independent emitters each; see the masks at the top. The bar within each mask represents the Rayleigh limit d R . The depicted areas in the final images differ from top to bottom, as the emitters are distributed over a smaller area. However, the areas are not shrunk according to relative distances, as the Airy disk’s size in the intensity measurements G ( 1 ) ( r ) remains the same for each run.

Fig. 5.
Fig. 5.

Imaging the same 3 × 3 array as used in Fig. 4, with grid constant d = 0.29 d R . The upper illustrations (a)–(e) show the resulting final images in real space, and the lower illustrations (f)–(i) show the corresponding reciprocal space for the images in (a)–(c) and (e). For image (d), the reciprocal space distribution of image (h) was used, however, with a reconstruction strategy different from the one used in (c). The ordinary intensity measurement G ( 1 ) ( r ) is shown in (a), CM 3 ( r ) in (b), and SIQCM 3 ( r ) with three different reconstruction approaches in (c)–(e). In (c), a homogenous disk was used for the reconstruction. In (d) the same disk was used, though negative values were cropped in the real-space distribution before taking the modulus. In (e), a triangular apodization was applied to the homogenous disk, the same procedure that was utilized for SIQCM 2 ( r ) in Fig. 4.

Fig. 6.
Fig. 6.

Imaging the same 3 × 3 array as used in Fig. 4, with grid constant d = 0.29 d R and added noise Δ . The illustrations show G ( 1 ) ( r ) , CM 2 ( r ) , SIM, and SIQCM 2 ( r ) (from left to right) and noise levels Δ = 0.1 , Δ = 0.3 , and Δ = 1.0 (from top to bottom), respectively.

Tables (1)

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Table 1. Coefficients of Higher Harmonics Relative to the Constant Term 0 k 0 r for Correlation Orders m = 1 , , 5

Equations (8)

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I 1 2 I 0 1 + I 0 .
I ( r ) = E ^ ( ) ( r ) E ^ ( + ) ( r ) i = 1 N h ( r r i ) ,
CM 2 ( r ) = ( G ( 1 ) ( r ) ) 2 1 2 G ( 2 ) ( r ) = i = 1 N h 2 ( r r i ) ,
G ( 1 ) ( r ) = i = 1 N [ h ( r r i ) × I str ( r i , α , φ ) ] ,
FT { h ( r ) * [ n ( r ) × I str ( r , φ , α ) ] } = H ( k ) × [ 1 2 n ˜ ( k ) + 1 4 e i φ n ˜ ( k k 0 ) + 1 4 e i φ n ˜ ( k k 0 ) ] ,
I l b l cos ( l k 0 r ) .
SIQCM 2 ( r ) = i = 1 N [ h 2 ( r r i ) × ( I str ( r i , α , φ ) ) 2 ] = h 2 ( r ) * [ n ( r ) × ( I str ( r , α , φ ) ) 2 ] ,
H 2 ( k ) × [ 3 8 n ˜ ( k ) + 1 4 e i φ n ˜ ( k k 0 ) + 1 4 e i φ n ˜ ( k + k 0 ) + 1 16 e 2 i φ n ˜ ( k 2 k 0 ) + 1 16 e 2 i φ n ˜ ( k + 2 k 0 ) ] ,

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