Abstract

Holographic speckle is a major impediment for the emerging applications of multiphoton holographic projection in biomedical imaging, photo-stimulation and micromachining. Time averaging of multiple shifted versions of a single hologram (“shift-averaging”) is a computationally-efficient method that was recently shown to deterministically eliminate holographic speckle in single-photon applications. Here, we extend these results and show, computationally and experimentally, that in two-photon holographic excitation shift-averaging also reduces holographic speckle better than “random” averaging of multiple calculated holograms.

© 2011 OSA

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References

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  1. N. J. Jenness, K. D. Wulff, M. S. Johannes, M. J. Padgett, D. G. Cole, and R. L. Clark, “Three-dimensional parallel holographic micropatterning using a spatial light modulator,” Opt. Express 16(20), 15942–15948 (2008).
    [CrossRef] [PubMed]
  2. E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72(3), 1810–1816 (2001).
    [CrossRef]
  3. C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods 5(9), 821–827 (2008).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. V. R. Daria, C. Stricker, R. Bowman, S. Redman, and H. A. Bachor, “Arbitrary multisite two-photon excitation in four dimensions,” Appl. Phys. Lett. 95(9), 093701 (2009).
    [CrossRef]
  15. G. Bautista, M. J. Romero, G. Tapang, and V. R. Daria, “Parallel two-photon photopolymerization of microgear patterns,” Opt. Commun. 282(18), 3746–3750 (2009).
    [CrossRef]
  16. E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2011

2010

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

S. Shoham, “Optogenetics meets optical wavefront shaping,” Nat. Methods 7(10), 798–799 (2010).
[CrossRef] [PubMed]

2009

J. Chen, S. M. Morris, T. D. Wilkinson, J. P. Freeman, and H. J. Coles, “High speed liquid crystal over silicon display based on the flexoelectro-optic effect,” Opt. Express 17(9), 7130–7137 (2009).
[CrossRef] [PubMed]

V. R. Daria, C. Stricker, R. Bowman, S. Redman, and H. A. Bachor, “Arbitrary multisite two-photon excitation in four dimensions,” Appl. Phys. Lett. 95(9), 093701 (2009).
[CrossRef]

G. Bautista, M. J. Romero, G. Tapang, and V. R. Daria, “Parallel two-photon photopolymerization of microgear patterns,” Opt. Commun. 282(18), 3746–3750 (2009).
[CrossRef]

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng. 6(6), 066004 (2009).
[CrossRef] [PubMed]

L. Golan and S. Shoham, “Speckle elimination using shift-averaging in high-rate holographic projection,” Opt. Express 17(3), 1330–1339 (2009).
[CrossRef] [PubMed]

Y. Pan, X. Xu, S. Solanki, X. Liang, R. B. A. Tanjung, C. Tan, and T.-C. Chong, “Fast CGH computation using S-LUT on GPU,” Opt. Express 17(21), 18543–18555 (2009).
[CrossRef] [PubMed]

2008

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

N. J. Jenness, K. D. Wulff, M. S. Johannes, M. J. Padgett, D. G. Cole, and R. L. Clark, “Three-dimensional parallel holographic micropatterning using a spatial light modulator,” Opt. Express 16(20), 15942–15948 (2008).
[CrossRef] [PubMed]

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits 2, 5 (2008).
[CrossRef] [PubMed]

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods 5(9), 821–827 (2008).
[CrossRef] [PubMed]

2001

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72(3), 1810–1816 (2001).
[CrossRef]

1995

1976

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Amako, J.

Anselmi, F.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Araya, R.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits 2, 5 (2008).
[CrossRef] [PubMed]

Bachor, H. A.

V. R. Daria, C. Stricker, R. Bowman, S. Redman, and H. A. Bachor, “Arbitrary multisite two-photon excitation in four dimensions,” Appl. Phys. Lett. 95(9), 093701 (2009).
[CrossRef]

Bautista, G.

G. Bautista, M. J. Romero, G. Tapang, and V. R. Daria, “Parallel two-photon photopolymerization of microgear patterns,” Opt. Commun. 282(18), 3746–3750 (2009).
[CrossRef]

Beck, R. J.

Bègue, A.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Bowman, R.

V. R. Daria, C. Stricker, R. Bowman, S. Redman, and H. A. Bachor, “Arbitrary multisite two-photon excitation in four dimensions,” Appl. Phys. Lett. 95(9), 093701 (2009).
[CrossRef]

Charpak, S.

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods 5(9), 821–827 (2008).
[CrossRef] [PubMed]

Chen, J.

Chong, T.-C.

Clark, R. L.

Cole, D. G.

Coles, H. J.

Daria, V. R.

V. R. Daria, C. Stricker, R. Bowman, S. Redman, and H. A. Bachor, “Arbitrary multisite two-photon excitation in four dimensions,” Appl. Phys. Lett. 95(9), 093701 (2009).
[CrossRef]

G. Bautista, M. J. Romero, G. Tapang, and V. R. Daria, “Parallel two-photon photopolymerization of microgear patterns,” Opt. Commun. 282(18), 3746–3750 (2009).
[CrossRef]

de Sars, V.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

Dearing, M. T.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72(3), 1810–1816 (2001).
[CrossRef]

DeSars, V.

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods 5(9), 821–827 (2008).
[CrossRef] [PubMed]

DiGregorio, D. A.

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods 5(9), 821–827 (2008).
[CrossRef] [PubMed]

Dufresne, E. R.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72(3), 1810–1816 (2001).
[CrossRef]

Emiliani, V.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods 5(9), 821–827 (2008).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

Farah, N.

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng. 6(6), 066004 (2009).
[CrossRef] [PubMed]

Freeman, J. P.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Glückstad, J.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Golan, L.

L. Golan and S. Shoham, “Speckle elimination using shift-averaging in high-rate holographic projection,” Opt. Express 17(3), 1330–1339 (2009).
[CrossRef] [PubMed]

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng. 6(6), 066004 (2009).
[CrossRef] [PubMed]

Goodman, J. W.

Grier, D. G.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72(3), 1810–1816 (2001).
[CrossRef]

Hand, D. P.

Isacoff, E. Y.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

Jenness, N. J.

Johannes, M. S.

Liang, X.

Lutz, C.

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods 5(9), 821–827 (2008).
[CrossRef] [PubMed]

Miura, H.

Morris, S. M.

Nikolenko, V.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits 2, 5 (2008).
[CrossRef] [PubMed]

Oron, D.

Otis, T. S.

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods 5(9), 821–827 (2008).
[CrossRef] [PubMed]

Padgett, M. J.

Pan, Y.

Papagiakoumou, E.

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

E. Papagiakoumou, V. de Sars, D. Oron, and V. Emiliani, “Patterned two-photon illumination by spatiotemporal shaping of ultrashort pulses,” Opt. Express 16(26), 22039–22047 (2008).
[CrossRef] [PubMed]

Parry, J. P.

Peterka, D. S.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits 2, 5 (2008).
[CrossRef] [PubMed]

Redman, S.

V. R. Daria, C. Stricker, R. Bowman, S. Redman, and H. A. Bachor, “Arbitrary multisite two-photon excitation in four dimensions,” Appl. Phys. Lett. 95(9), 093701 (2009).
[CrossRef]

Reutsky, I.

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng. 6(6), 066004 (2009).
[CrossRef] [PubMed]

Romero, M. J.

G. Bautista, M. J. Romero, G. Tapang, and V. R. Daria, “Parallel two-photon photopolymerization of microgear patterns,” Opt. Commun. 282(18), 3746–3750 (2009).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Sheets, S. A.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72(3), 1810–1816 (2001).
[CrossRef]

Shephard, J. D.

Shoham, S.

S. Shoham, “Optogenetics meets optical wavefront shaping,” Nat. Methods 7(10), 798–799 (2010).
[CrossRef] [PubMed]

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng. 6(6), 066004 (2009).
[CrossRef] [PubMed]

L. Golan and S. Shoham, “Speckle elimination using shift-averaging in high-rate holographic projection,” Opt. Express 17(3), 1330–1339 (2009).
[CrossRef] [PubMed]

Solanki, S.

Sonehara, T.

Spalding, G. C.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72(3), 1810–1816 (2001).
[CrossRef]

Stricker, C.

V. R. Daria, C. Stricker, R. Bowman, S. Redman, and H. A. Bachor, “Arbitrary multisite two-photon excitation in four dimensions,” Appl. Phys. Lett. 95(9), 093701 (2009).
[CrossRef]

Tan, C.

Tanjung, R. B. A.

Tapang, G.

G. Bautista, M. J. Romero, G. Tapang, and V. R. Daria, “Parallel two-photon photopolymerization of microgear patterns,” Opt. Commun. 282(18), 3746–3750 (2009).
[CrossRef]

Watson, B. O.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits 2, 5 (2008).
[CrossRef] [PubMed]

Wilkinson, T. D.

Woodruff, A.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits 2, 5 (2008).
[CrossRef] [PubMed]

Wulff, K. D.

Xu, X.

Yuste, R.

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits 2, 5 (2008).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

V. R. Daria, C. Stricker, R. Bowman, S. Redman, and H. A. Bachor, “Arbitrary multisite two-photon excitation in four dimensions,” Appl. Phys. Lett. 95(9), 093701 (2009).
[CrossRef]

Front. Neural Circuits

V. Nikolenko, B. O. Watson, R. Araya, A. Woodruff, D. S. Peterka, and R. Yuste, “SLM microscopy: scanless two-photon imaging and photostimulation using spatial light modulators,” Front. Neural Circuits 2, 5 (2008).
[CrossRef] [PubMed]

J. Neural Eng.

L. Golan, I. Reutsky, N. Farah, and S. Shoham, “Design and characteristics of holographic neural photo-stimulation systems,” J. Neural Eng. 6(6), 066004 (2009).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

Nat. Methods

C. Lutz, T. S. Otis, V. DeSars, S. Charpak, D. A. DiGregorio, and V. Emiliani, “Holographic photolysis of caged neurotransmitters,” Nat. Methods 5(9), 821–827 (2008).
[CrossRef] [PubMed]

E. Papagiakoumou, F. Anselmi, A. Bègue, V. de Sars, J. Glückstad, E. Y. Isacoff, and V. Emiliani, “Scanless two-photon excitation of channelrhodopsin-2,” Nat. Methods 7(10), 848–854 (2010).
[CrossRef] [PubMed]

S. Shoham, “Optogenetics meets optical wavefront shaping,” Nat. Methods 7(10), 798–799 (2010).
[CrossRef] [PubMed]

Opt. Commun.

G. Bautista, M. J. Romero, G. Tapang, and V. R. Daria, “Parallel two-photon photopolymerization of microgear patterns,” Opt. Commun. 282(18), 3746–3750 (2009).
[CrossRef]

Opt. Express

Optik (Jena)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Rev. Sci. Instrum.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instrum. 72(3), 1810–1816 (2001).
[CrossRef]

Other

T. S. McKechnie, “Speckle Reduction,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, 1975).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

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Figures (5)

Fig. 1
Fig. 1

(a) Illustration of the c by c neighborhood of an arbitrary point in the (u,v) plane. (b) Illustration of two-dimensional cyclic shifts. Red ovals indicate a reference area in the hologram.

Fig. 2
Fig. 2

Fraction of terms that remain out of c8 initial terms after shift averaging.

Fig. 3
Fig. 3

Simulation of the performance of the shift-averaging method for one-photon and two-photon phenomena. (a) Performance for a single square patch (left panes) and a number of small patches (right panes). (b) Comparison of speckle contrast reduction for one- and two-photon shift-averaging and two-photon regular averaging as a function of the number of shifts in each direction, c.

Fig. 4
Fig. 4

Sketch of the optical system. BE – Beam Expander; λ/2 – half-wave plate; SLM – Spatial Light Modulator; M – mirror; Lenses L1 and L2 form a telescope, while L3 is used to image the plane onto the CCD; DM – dichroic mirror; OL – Objective Lens; B – zero-order point blocker.

Fig. 5
Fig. 5

(a) Demonstration of shift averaging in two-photon fluorescence. Using 16 holograms (4x4 shift averaging) the speckle contrast was reduced almost tenfold. Scale bar is 10µm. (b) Comparison of the cross-section along the diameters of the different patches before (left) and after (right) shift-averaging.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

F mn =exp(i ϕ mn );m,n=1,2,..,M,
f kl = m,n=0 M1 exp( i ϕ mn )exp[ i2π( mk M + nl M ) ] = I kl exp( i ψ kl ).
t(x,y)=rect(x,y){ [ m,n=1 M exp(i ϕ mn )δ(x m M ,y n M ) ]rect( x M , y M ) },
E(u,v)=(t(x,y))= k= l= f kl S kl (u,v) ,
S kl (u,v)=sinc(uk,vl)sinc( u M , v M );
E( u,v ) k=1 c l=1 c f kl S kl ( u,v ) ,
I( u,v )= | E( u,v ) | 2 =E( u,v ) E * ( u,v )= k=1 c l=1 c r=1 c s=1 c f kl f rs * S kl S rs ,
I(u,v) = k=1 c k=1 c k=1 c k=1 c S kl S rs ( 1 N a=1 N f kl a f rs a,* ),
1 N a=1 N f kl a f rs a,* = δ kr δ ls | f kl | 2 ,
I(u,v) ideal = k=1 c l=1 c S kl 2 f kl f kl * = k=1 c l=1 c S kl 2 I kl .
f kl a = f kl exp[ i2π( k d 1 a M + l d 2 a M ) ],
N= c 2 , d 1 ab =a M c , d 2 ab =b M c ,
1 c 2 a=1 c b=1 c f kl a f rs a,* = 1 c 2 f kl f rs * ( a=1 c exp[ i2π( kr ) a c ] )( b=1 c exp[ i2π( ls ) b c ] ) = δ k,r δ l,s | f kl | 2 ,
I 2 ( u,v )= | E( u,v ) | 4 =( k=1 c l=1 c r=1 c s=1 c f kl f rs * S kl S rs )( k=1 c l=1 c r=1 c s=1 c f kl f rs * S kl S rs ) = k=1 c l=1 c r=1 c s=1 c g=1 c h=1 c p=1 c q=1 c f kl f rs * f gh f pq * S kl S rs S gh S pq ,
I 2 ( u,v ) = k=1 c l=1 c r=1 c s=1 c g=1 c h=1 c p=1 c q=1 c S kl S rs S gh S pq ( 1 N a=1 N f kl a f rs a,* f gh a f pq a,* ) .
1 N a=1 N f kl a f rs a,* f gh a f pq a,* = δ krgp δ lshq | f kl | 4 ,
I 2 ( u,v ) ideal = k=1 c l=1 c S kl 4 f kl 2 f kl 2* = k=1 c l=1 c S kl 4 I kl 2 .
1 c 2 a=1 c b=1 c f kl ab f rs ab,* f gh ab f pq ab,* = 1 c 2 f kl f rs * f gh f pq * a=1 c b=1 c exp [ i2π( kr+gp ) a c ]exp[ i2π( ls+hq ) b c ] = δ k+g,r+p δ l+h,s+q | f kl | 4
{ ( k+g=r+p ) ( l+h=s+q ) .
( 2 c 3 +c ) 2 /9 c 2 c 8 = 4 9 c 2 + 4 9 c 4 8 9 c 6 ,
C= σ G G

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