Abstract

High-content biological microscopy targets high-resolution imaging across large fields-of-view (FOVs). Recent works have demonstrated that computational imaging can provide efficient solutions for high-content microscopy. Here, we use speckle structured illumination microscopy (SIM) as a robust and cost-effective solution for high-content fluorescence microscopy with simultaneous high-content quantitative phase (QP). This multi-modal compatibility is essential for studies requiring cross-correlative biological analysis. Our method uses laterally-translated Scotch tape to generate high-resolution speckle illumination patterns across a large FOV. Custom optimization algorithms then jointly reconstruct the sample’s super-resolution fluorescent (incoherent) and QP (coherent) distributions, while digitally correcting for system imperfections such as unknown speckle illumination patterns, system aberrations and pattern translations. Beyond previous linear SIM works, we achieve resolution gains of 4× the objective’s diffraction-limited native resolution, resulting in 700 nm fluorescence and 1.2 μm QP resolution, across a FOV of 2×2.7 mm 2, giving a space-bandwidth product (SBP) of 60 megapixels.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (5)

2016 (1)

2015 (6)

H. Yilmaz, E. G. V. Putten, J. Bertolotti, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Speckle correlation resolution enhancement of wide-field fluorescence imaging,” Optica 2, 424–429 (2015).
[Crossref]

A. Orth, M. J. Tomaszewski, R. N. Ghosh, and E. Schonbrun, “Gigapixel multispectral microscopy,” Optica 2, 654–662 (2015).
[Crossref]

L. Tian, Z. Liu, L. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349, aab3500 (2015).
[Crossref]

A. Jost, E. Tolstik, P. Feldmann, K. Wicker, A. Sentenac, and R. Heintzmann, “Optical sectioning and high resolution in single-slice structured illumination microscopy by thick slice blind-SIM reconstruction,” PLoS ONE 10, e0132174 (2015).
[Crossref] [PubMed]

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33213–33238 (2015).
[Crossref]

2014 (4)

2013 (10)

A. Orth and K. Crozier, “Gigapixel fluorescence microscopy with a water immersion microlens array,” Opt. Express 21, 2361–2368 (2013).
[Crossref] [PubMed]

P. Gao, G. Pedrini, and W. Osten, “Structured illumination for resolution enhancement and autofocusing in digital holographic microscopy,” Opt. Lett. 38, 1328–1330 (2013).
[Crossref] [PubMed]

S. Pang, C. Han, J. Erath, A. Rodriguez, and C. Yang, “Wide field-of-view Talbot grid-based microscopy for multicolor fluorescence imaging,” Opt. Express 21, 14555–14565 (2013).
[Crossref] [PubMed]

S. Chowdhury and J. A. Izatt, “Structured illumination quantitative phase microscopy for enhanced resolution amplitude and phase imaging,” Biomed. Opt. Express 4, 1795–1805 (2013).
[Crossref] [PubMed]

P. von Olshausen and A. Rohrbach, “Coherent total internal reflection dark-field microscopy: label-free imaging beyond the diffraction limit,” Opt. Lett. 38, 4066–4069 (2013).
[Crossref] [PubMed]

R. Ayuk, H. Giovannini, A. Jost, E. Mudry, J. Girard, T. Mangeat, N. Sandeau, R. Heintzmann, K. Wicker, K. Belkebir, and A. Sentenac, “Structured illumination fluorescence microscopy with distorted excitations using a filtered blind-SIM algorithm,” Opt. Lett. 38, 4723–4726 (2013).
[Crossref] [PubMed]

J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Scientific Reports 3, 2075: 1–6 (2013).
[Crossref]

D. Dan, M. Lei, B. Yao, W. Wang, M. Winterhalder, A. Zumbusch, Y. Qi, L. Xia, S. Yan, Y. Yang, P. Gao, T. Ye, and W. Zhao, “DMD-based LED-illumination Super-resolution and optical sectioning microscopy,” Scientific Reports 3, 1116 (2013).
[Crossref] [PubMed]

A. Greenbaum, W. Luo, B. Khademhosseinieh, T.-W. Su, A. F. Coskun, and A. Ozcan, “Increased space-bandwidth product in pixel super-resolved lensfree on-chip microscopy,” Scientific reports 3: 1717 (2013).
[Crossref]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photon. 7, 739–745 (2013).
[Crossref]

2012 (4)

2011 (1)

2010 (1)

2009 (1)

F. R. Dee, “Virtual microscopy in pathology education,” Human Pathol 40, 1112–1121 (2009).
[Crossref]

2008 (2)

M. H. Kim, Y. Park, D. Seo, Y. J. Lim, D.-I. Kim, C. W. Kim, and W. H. Kim, “Virtual microscopy as a practical alternative to conventional microscopyin pathology education,” Basic Appl. Pathol. 1, 46–48 (2008).
[Crossref]

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

2007 (2)

V. Laketa, J. C. Simpson, S. Bechtel, S. Wiemann, and R. Pepperkok, “High-content microscopy identifies new neurite outgrowth regulators,” Mol. Biol. Cell 18, 242–252 (2007).
[Crossref]

V. Starkuviene and R. Pepperkok, “The potential of high-content high-throughput microscopy in drug discovery,” Br. J. Pharmacol 152, 62–71 (2007).
[PubMed]

2006 (4)

A. Trounson, “The production and directed differentiation of human embryonic stem cells,” Endocr. Rev. 27(2), 208–219 (2006).
[Crossref] [PubMed]

R. Pepperkok and J. Ellenberg, “High-throughput fluorescence microscopy for systems biology,” Nat. Rev. Mol. Cell Biol. 7, 690–696 (2006).
[Crossref] [PubMed]

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] [PubMed]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nature Methods 3, 793–795 (2006).
[Crossref] [PubMed]

2005 (2)

M. G. L. Gustafson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” PNAS 102, 13081–13086 (2005).
[Crossref]

J. C. Yarrow, G. Totsukawa, G. T. Charras, and T. J. Mitchison, “Screening for cell migration inhibitors via automated microscopy reveals a Rho-kinase Inhibitor,” Chem. Biol. 12, 385–395 (2005).
[Crossref] [PubMed]

2004 (2)

B. Mccullough, X. Ying, T. Monticello, and M. Bonnefoi, “Digital microscopy imaging and new approaches in toxicologic pathology,” Toxicol Pathol. 32 (suppl 2), 49–58 (2004).
[Crossref] [PubMed]

U. S. Eggert, A. A. Kiger, C. Richter, Z. E. Perlman, N. Perrimon, T. J. Mitchison, and C. M. Field, “Parallel chemical genetic and genome-wide RNAi screens identify cytokinesis inhibitors and targets,” PLoS Biol. 2, e379 (2004).
[Crossref] [PubMed]

2003 (1)

2001 (1)

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” PNAS 98, 11301–11305 (2001).
[Crossref] [PubMed]

2000 (1)

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” Journal of Microscopy 198, 82–87 (2000).
[Crossref] [PubMed]

1999 (1)

R. Heintzmann and C. Cremer, “Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating,” Proc. SPIE 3568, 185–196 (1999).
[Crossref]

1996 (1)

1994 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[Crossref]

1983 (1)

Y. Nesterov, “A method for solving the convex programming problem with convergence rate O(1/k2),” Dokl. Akad. Nauk SSSR 269, 543–547 (1983).

1982 (1)

1971 (1)

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

1967 (1)

Allain, M.

A. Negash, S. Labouesse, N. Sandeau, M. Allain, H. Giovannini, J. Idier, R. Heintzmann, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Improving the axial and lateral resolution of three-dimensional fluorescence microscopy using random speckle illuminations,” J. Opt. Soc. Am. A 33, 1089–1094 (2016).
[Crossref]

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. L. Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photon. 6, 312–315 (2012).
[Crossref]

S. Labouesse, M. Allain, J. Idier, S. Bourguignon, A. Negash, P. Liu, and A. Sentenac, “Joint reconstruction strategy for structured illumination microscopy with unknown illuminations,” ArXiv: 1607.01980 (2016).

Ayuk, R.

Baird, M. A.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349, aab3500 (2015).
[Crossref]

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nature Methods 3, 793–795 (2006).
[Crossref] [PubMed]

Beach, J. R.

D. Li, L. Shao, B.-C. Chen, X. Zhang, M. Zhang, B. Moses, D. E. Milkie, J. R. Beach, J. A. Hammer, M. Pasham, T. Kirchhausen, M. A. Baird, M. W. Davidson, P. Xu, and E. Betzig, “Extended-resolution structured illumination imaging of endocytic and cytoskeletal dynamics,” Science 349, aab3500 (2015).
[Crossref]

Bechtel, S.

V. Laketa, J. C. Simpson, S. Bechtel, S. Wiemann, and R. Pepperkok, “High-content microscopy identifies new neurite outgrowth regulators,” Mol. Biol. Cell 18, 242–252 (2007).
[Crossref]

Belkebir, K.

Bertolotti, J.

Betzig, E.

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Richter, C.

U. S. Eggert, A. A. Kiger, C. Richter, Z. E. Perlman, N. Perrimon, T. J. Mitchison, and C. M. Field, “Parallel chemical genetic and genome-wide RNAi screens identify cytokinesis inhibitors and targets,” PLoS Biol. 2, e379 (2004).
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M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nature Methods 3, 793–795 (2006).
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J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, “Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,” Scientific Reports 3, 2075: 1–6 (2013).
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W. Bishara, T.-W. Su, A. F. Coskun, and A. Ozcan, “Lensfree on-chip microscopy over a wide field-of-view using pixel super-resolution,” Opt. Express 18, 11181–11191 (2010).
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J. C. Yarrow, G. Totsukawa, G. T. Charras, and T. J. Mitchison, “Screening for cell migration inhibitors via automated microscopy reveals a Rho-kinase Inhibitor,” Chem. Biol. 12, 385–395 (2005).
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Zhang, M.

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Zhao, W.

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Zhong, J.

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Basic Appl. Pathol. (1)

M. H. Kim, Y. Park, D. Seo, Y. J. Lim, D.-I. Kim, C. W. Kim, and W. H. Kim, “Virtual microscopy as a practical alternative to conventional microscopyin pathology education,” Basic Appl. Pathol. 1, 46–48 (2008).
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S. Chowdhury and J. A. Izatt, “Structured illumination quantitative phase microscopy for enhanced resolution amplitude and phase imaging,” Biomed. Opt. Express 4, 1795–1805 (2013).
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J. C. Yarrow, G. Totsukawa, G. T. Charras, and T. J. Mitchison, “Screening for cell migration inhibitors via automated microscopy reveals a Rho-kinase Inhibitor,” Chem. Biol. 12, 385–395 (2005).
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A. Jost, E. Tolstik, P. Feldmann, K. Wicker, A. Sentenac, and R. Heintzmann, “Optical sectioning and high resolution in single-slice structured illumination microscopy by thick slice blind-SIM reconstruction,” PLoS ONE 10, e0132174 (2015).
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A. Greenbaum, W. Luo, B. Khademhosseinieh, T.-W. Su, A. F. Coskun, and A. Ozcan, “Increased space-bandwidth product in pixel super-resolved lensfree on-chip microscopy,” Scientific reports 3: 1717 (2013).
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Supplementary Material (3)

NameDescription
» Visualization 1       This is a 2D structured illumination microscopy fluorescence reconstruction of 1um beads using random speckle illumination. The full field of view is 2.7x3.3 mm^2 and the resolution is around 700 nm.
» Visualization 2       This is the full field of view fluorescence channel reconstruction of the computational structured illumination microscopy with 2um beads using random speckle illumination. The field of view is around 2x2.7 mm^2 and the resolution is around 700 nm.
» Visualization 3       This is the full field of view phase reconstruction of the computational structured illumination microscopy with 2um beads using random speckle illumination. The field of view is around 2x2.7 mm^2 and the resolution is around 1.23 um.

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Figures (7)

Fig. 1
Fig. 1 Structured illumination microscopy (SIM) with laterally-translated Scotch tape as the patterning element, achieving 4× resolution gain. Our imaging system has both an incoherent arm, where Sensor-F captures raw fluorescence images (at the emission wavelength, λ em = 605 nm) for fluorescence super-resolution, and a coherent arm, where Sensor-C1 and Sensor-C2 capture images with different defocus (at the laser illumination wavelength, λ ex = 532 nm) for both super-resolution phase reconstruction and speckle trajectory calibration. OBJ: objective, AP: adjustable iris-aperture, DM: dichroic mirror, SF: spectral filter, ND-F: neutral-density filter.
Fig. 2
Fig. 2 Verification of fluorescence super-resolution with 4× resolution gain. Widefield images, for comparison, were acquired at (a) 0.1 NA and (e) 0.4 NA by adjusting the aperture size. (b) The Scotch tape speckle pattern creates much higher spatial frequencies (∼0.35 NA) than the 0.1 NA detection system can measure. (c) Using the 0.1 NA aperture, we acquire low-resolution fluorescence images for different lateral positions of the Scotch tape. (d) The reconstructed SIM image contains spatial frequencies up to ∼0.4 NA and is in agreement with (e) the deconvolved widefield image with the system operating at 0.4 NA.
Fig. 3
Fig. 3 Verification of coherent quantitative phase (QP) super-resolution with 4× resolution gain. (a) Low-resolution intensity image and (b) “ground truth” phase at NA=0.4, for comparison. (c) Raw acquisitions of the speckle-illuminated sample intensity from two focus planes, collected with 0.1 NA. (d) Reconstructed SR amplitude and QP, demonstrating 4× resolution gain.
Fig. 4
Fig. 4 Reconstructed super-resolution fluorescence with 4× resolution gain across the full FOV (See Visualization 1). Four zoom-ins of regions-of-interest (ROIs) are compared to their widefield counterparts.
Fig. 5
Fig. 5 Reconstructed multimodal (fluorescence and quantitative phase) high-content imaging (See Visualization 2 and Visualization 3). Zoom-ins for three ROIs compare the widefield, super-resolved fluorescence, coherent intensity, and super-resolved phase reconstructions.
Fig. 6
Fig. 6 Algorithmic self-calibration significantly improves fluorescence super-resolution reconstructions. Here, we compare the resconstructed fluorescence image, speckle intensity, and OTF with no correction, OTF correction, and both OTF correction and scanning position correction. The right panel shows the overlay of the uncorrected and corrected scanning position trajectories.
Fig. 7
Fig. 7 Algorithmic self-calibration significantly improves coherent super-resolution reconstructions. We show a comparison of reconstructed amplitude, phase, speckle amplitude, and phase of the pupil function with no correction, pupil correction, and both pupil correction and scanning position correction. The right panel shows the overlay of scannning position trajectory for the in-focus and defocused cameras before and after correction.

Tables (2)

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Algorithm 1 Fluorescence imaging reconstruction

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Algorithm 2 Coherent imaging reconstruction

Equations (25)

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I f , l ( r ) = [ o f ( r ) C { p f ( r r l ) } ] h f ( r ) , l = 1 , , N img ,
min o f , p f , h f , r 1 , , r N img f f ( o f , p f , h f , r 1 , , r N img ) = l = 1 N img f f , l ( o f , p f , h f , r l ) , where    f f , l ( o f , p f , h f , r l ) = r | I f , l ( r ) [ o f ( r ) C { p f ( r r l ) } ] h f ( r ) | 2 .
I c , l z ( r ) = | [ o c ( r ) C { p c ( r r l ) } ] h c , z ( r ) | 2 = | g c , l z ( r ) | 2 , l = 1 , , N img , ; z = z 0 , z 1 ,
minimize o c , p c , h c , r 1 z 0 , r 1 z 1 , , r N img z 0 , r N img z 1 f c ( o c , p c , h c , r 1 z 0 , r 1 z 1 , , r N img z 0 , r N img z 1 ) = l , z f c , l z ( o c , p c , h c , r l z ) , where    f c , l z ( o c , p c , h c , r l z ) = r | I c , l z ( r ) | [ o c ( r ) C { p c ( r r l z ) } ] h c , z ( r ) | | 2 .
r l z = R [ I c , 1 z ( r ) , I c , l z ( r ) ] ,
I f , l = H f diag ( S ( r l ) p f ) o f ,
H f = F M 1 diag ( h ˜ f ) F M , S ( r l ) = Q F N 1 diag ( e ( r l ) ) F N ,
f f , l ( o f , p f , h ˜ f , r l ) = f f , l T f f , l = I f , l H f diag ( S ( r l ) p f ) o f 2 2 ,
I c , l z = | g c , l z | 2 ,
g c , l z = H c , z diag ( S ( r l z ) p c ) o c H c , z = F M 1 diag ( h ˜ c ) diag ( h ˜ z ) F M .
f c , l z ( o c , p c , h ˜ c , r l z ) = f c , l z T f c , l z = I c , l z | g c , l z | 2 2 ,
f f , l o f = ( f f , l f f , l ) ( f f , l o f ) = ( 2 f f , l T ) f t ( H f diag ( S ( r l ) p f ) .
o f f f , l = ( f f , l o f ) T = 2 diag ( S ( r l ) p f ) H f T f f , l .
f f , l = I f , l H f diag ( o ) S ( r l ) p f .
f f , l p f = ( f f , l f f , l ) ( f f , l p f ) = ( 2 f f , l T ) ( H f diag ( o f ) S ( r l ) ) p f f f , l = ( f f , l p f ) T = 2 S ( r l ) T diag ( o f ) H f T f f , l .
f f , l = I f , l F M 1 diag ( F M diag ( S ( r l ) p f o f ) ) h ˜ f .
f f , l h ˜ f = ( f f , l f f , l ) ( f f , l h ˜ f ) = ( 2 f f , l T ) ( F M 1 diag ( F M diag ( S ( r l ) p f o f ) ) )   h ˜ f f f , l = ( f f , l h ˜ f ) = 2 diag ( F M diag ( S ( r l ) p f o f ) ¯ ) F M f f , l ,
f f , l = I l H f diag ( o f ) Q F N 1 diag ( F N p f ) e ( r l ) .
f f , l q l = ( f f , l f f , l ) ( f f , l e ( r l ) ) ( e ( r l ) q l ) = ( 2 f f , l T ) ( H f diag ( o f ) Q F N 1 diag ( F N p f ) ) ( diag ( j 2 π u q ) e ( r l ) ) ,
o f f f , l ( o f , p f , h f , r l ) = 2 p f ( r r l ) [ h f * ( r ) ( I f , l ( r ) [ o f ( r ) C { p f ( r r l ) } ] h f ( r ) ) ] , p f f f , l ( o f , p f , h f , r l ) = 2 δ ( r + r l ) P { o f ( r ) [ h f * ( r ) ( I f , l ( r ) [ o f ( r ) C { p f ( r r l ) } ] h f ( r ) ) ] } , h ˜ f f f , l ( o f , p f , h f , r l ) = 2 ( F { o f ( r ) C { p f ( r r l ) } } ) * F { I f , l ( r ) [ o f ( r ) C { p f ( r r l ) } ] h f ( r ) } , q l f f , l ( o f , p f , h f , r l ) = 2 { r ( I f , l ( r ) [ o f ( r ) C { p f ( r r l ) } ] h f ( r ) ) h f ( r ) [ o f ( r ) C { p f ( r r l ) q l } ] } ,
f c , l z o c = ( f c , l z f c , l z ) ( f c , l z g c , l z ) ( g c , l z o c ) = ( 2 f c , l z T ) ( 1 2 diag ( g c , l z ¯ | g c , l z | ) ) ( H c , z diag ( S ( r l z ) ) p c ) o c f c , l z = ( f c , l z o c ) = diag ( S ( r l z ) p c ¯ ) H c , z diag ( g c , l z | g c , l z | ) f c , l z ,
f c , l z p c = ( f c , l z f c , l z ) ( f c , l z g c , l z ) ( g c , l z p c ) = ( 2 f c , l z T ) ( 1 2 diag ( g c , l z ¯ | g c , l z | ) ) ( H c , z diag ( o c ) S ( r l z ) ) p c f c , l z = ( f c , l z p c ) = S ( r l z ) diag ( o c ¯ ) H c , z diag ( g c , l z | g c , l z | ) f c , l z .
f c , l z h ˜ c = ( f c , l z f c , l z ) ( f c , l z g c , l z ) ( g c , l z h ˜ c ) = ( 2 f c , l z T ) ( 1 2 diag ( g c , l z ¯ | g c , l z | ) ) ( F M 1 diag [ F M diag ( S ( r l z ) p c ) o c ] diag ( h ˜ z ) ) h ˜ c f c , l z = ( f c , l z h ˜ c ) = diag ( h ˜ z ¯ ) diag [ F M diag ( S ( r l z ) p c ) o c ¯ ] F M diag ( g c , l z | g c , l z | ) f c , l z .
f c , l z q l z = ( f c , l z f c , l z ) [ ( f c , l z g c , l z ) ( g c , l z e ( r l z ) ) ( e ( r l z ) q l ) + ( f c , l z g c , l z ¯ ) ( g c , l z ¯ e ( r l z ) ¯ ) ( e ( r l z ) ¯ q l ) ] = 2 ( f c , l z f c , l z ) Re { ( f c , l z g c , l z ) ( g c , l z e ( r l z ) ) ( e ( r l z ) q l ) } = 2 ( 2 f c , l z T ) Re { ( 1 2 diag ( g c , l z ¯ | g c , l z | ) ) ( H c , z diag ( o c ) Q F N 1 diag ( F N p c ) ) ( diag ( j 2 π u q ) e ( r l z ) ) } = 2 Re { f c , l z T diag ( g c , l z ¯ | g c , l z | ) H c , z diag ( o c ) Q F N 1 diag ( F N p c ) diag ( j 2 π u q ) e ( r l z ) } ,
o c f c , l z ( o c , p c , h c , r l z ) = p c * ( r r l z ) [ h c , z * ( r ) ( ( I c , l z ( r ) | g c , l z ( r ) | 1 ) g c , l z ( r ) ) ] p c f c , l z ( o c , p c , h c , r l z ) = δ ( r + r l z ) P { o c * ( r ) [ h c , z * ( r ) ( ( I c , l z ( r ) | g c , l z ( r ) | 1 ) g c , l z ( r ) ) ] } h ˜ c f c , l z ( o c , p c , h c , r l z ) = h ˜ z * ( u ) F { p c ( r r l z ) o c ( r ) ) } * F { ( I c , l z ( r ) | g c , l z ( r ) | 1 ) g c , l z ( r ) } q l z f c , l z ( o c , p c , h c , r l z ) = 2 Re { r [ ( I c , l z ( r ) | g c , l z ( r ) | 1 ) g c , l z * ( r ) ] [ h c , z ( r ) ( o c ( r ) C { p c ( r r l z ) q l z } ) ] } .

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