Abstract

Coherence-gated wavefront sensing (CGWS) allows the determination of wavefront aberrations in strongly scattering tissue and their correction by adaptive optics. This allows, e.g., the restoration of the diffraction limit in light microscopy. Here, we develop a model, based on ray tracing of ballistic light scattered from a set of discrete scatterers, to characterize CGWS performance as it depends on coherence length, scatterer density, coherence-gate position, and polarization. The model is evaluated by using Monte Carlo simulation and verified against experimental measurements. We show, in particular, that all aberrations needed for adaptive wavefront restoration are correctly sensed if circularly polarized light is used.

© 2007 Optical Society of America

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2006 (2)

M. Rueckel, J. A. Mack-Bucher, and W. Denk, 'Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,' Proc. Natl. Acad. Sci. U.S.A. 103, 17137-17142 (2006).
[CrossRef] [PubMed]

M. Rueckel and W. Denk, 'Coherence-gated wavefront sensing using a virtual Shack-Hartmann sensor,' Proc. SPIE 6306, 63060H (2006).
[CrossRef]

2005 (4)

B. Karamata, M. Laubscher, M. Leutenegger, S. Bourquin, T. Lasser, and P. Lambelet, 'Multiple scattering in optical coherence tomography. I. Investigation and modeling,' J. Opt. Soc. Am. A 22, 1369-1379 (2005).
[CrossRef]

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, 'Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,' Microsc. Res. Tech. 67, 36-44 (2005).
[CrossRef] [PubMed]

M. J. Booth, M. Schwertner, and T. Wilson, 'Specimen-induced aberrations and adaptive optics for microscopy,' Proc. SPIE 5894, 589403 (2005).
[CrossRef]

J. D. Wilson and T. H. Foster, 'Mie theory interpretations of light scattering from intact cells,' Opt. Lett. 30, 2442-2444 (2005).
[CrossRef] [PubMed]

2004 (3)

2003 (3)

P. N. Marsh, D. Burns, and J. M. Girkin, 'Practical implementation of adaptive optics in multiphoton microscopy,' Opt. Express 11, 1123-1130 (2003).
[CrossRef] [PubMed]

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, 'Optical coherence tomography--principles and applications,' Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

R. Gens, 'Two-dimensional phase unwrapping for radar interferometry: developments and new challenges,' Int. J. Remote Sens. 24, 703-710 (2003).
[CrossRef]

2002 (4)

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Adaptive aberration correction in a confocal microscope,' Proc. Natl. Acad. Sci. U.S.A. 99, 5788-5792 (2002).
[CrossRef] [PubMed]

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, 'Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,' J. Microsc. 206, 65-71 (2002).
[CrossRef] [PubMed]

J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, 'Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures,' J. Biomed. Opt. 7, 378-387 (2002).
[CrossRef] [PubMed]

A. Rohrbach and E. H. K. Stelzer, 'Three-dimensional position detection of optically trapped dielectric particles,' J. Appl. Phys. 91, 5474-5488 (2002).
[CrossRef]

2000 (4)

K. Bahlmann and S. W. Hell, 'Depolarization by high aperture focusing,' Appl. Phys. Lett. 77, 612-614 (2000).
[CrossRef]

J. J. Zhu, J. A. Esteban, Y. Hayashi, and R. Malinow, 'Postnatal synaptic potentiation: Delivery of GluR4-containing AMPA receptors by spontaneous activity,' Nat. Neurosci. 3, 1098-1106 (2000).
[CrossRef] [PubMed]

M. A. A. Neil, R. Juskaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, 'Adaptive aberration correction in a two-photon microscope,' J. Microsc. 200, 105-108 (2000).
[CrossRef] [PubMed]

C. W. Chen and H. A. Zebker, 'Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,' J. Opt. Soc. Am. A 17, 401-414 (2000).
[CrossRef]

1999 (2)

1998 (3)

1997 (1)

1994 (1)

N. Shvartsman and I. Freund, 'Vortices in random wave-fields--nearest-neighbor anticorrelations,' Phys. Rev. Lett. 72, 1008-1011 (1994).
[CrossRef] [PubMed]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, 'Optical coherence tomography,' Science 254, 1178-1181 (1991).
[CrossRef]

1990 (1)

W. Denk, J. H. Strickler, and W. W. Webb, 'Two-photon laser scanning fluorescence microscopy,' Science 248, 73-76 (1990).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

1979 (1)

R. Cubalchini, 'Modal wavefront estimation from phase derivative measurements,' J. Opt. Soc. Am. 69, 973-977 (1979).
[CrossRef]

1978 (1)

J. W. Hardy, 'Active optics: a new technology for the control of light (review article),' Proc. IEEE 66, 651-697 (1978).
[CrossRef]

1976 (1)

1971 (1)

R. V. Shack and B. C. Platt, 'Abstract. Production and use of a lenticular Hartmann screen,' J. Opt. Soc. Am. 61, 656-660 (1971).

1957 (1)

S. Inoue and W. L. Hyde, 'Studies on depolarization of light at microscope lens surfaces. 2. The simultaneous realization of high resolution and high sensitivity with the polarizing microscope,' J. Biophys. Biochem. Cytol. 3, 831-838 (1957).
[CrossRef] [PubMed]

1941 (1)

1934 (1)

A. Khintchine, 'Correlation theories of stationary stochastic processes,' Math. Ann. 109, 604-615 (1934).
[CrossRef]

1930 (1)

N. Wiener, 'Generalized harmonic analysis,' Acta Math. 55, 117-258 (1930).
[CrossRef]

1908 (1)

G. Mie, 'Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,' Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Acta Math. (1)

N. Wiener, 'Generalized harmonic analysis,' Acta Math. 55, 117-258 (1930).
[CrossRef]

Ann. Phys. (1)

G. Mie, 'Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,' Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

K. Bahlmann and S. W. Hell, 'Depolarization by high aperture focusing,' Appl. Phys. Lett. 77, 612-614 (2000).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. M. Schmitt, 'Optical coherence tomography (OCT): A review,' IEEE J. Sel. Top. Quantum Electron. 5, 1205-1215 (1999).
[CrossRef]

Int. J. Remote Sens. (1)

R. Gens, 'Two-dimensional phase unwrapping for radar interferometry: developments and new challenges,' Int. J. Remote Sens. 24, 703-710 (2003).
[CrossRef]

J. Appl. Phys. (1)

A. Rohrbach and E. H. K. Stelzer, 'Three-dimensional position detection of optically trapped dielectric particles,' J. Appl. Phys. 91, 5474-5488 (2002).
[CrossRef]

J. Biomed. Opt. (1)

J. R. Mourant, T. M. Johnson, S. Carpenter, A. Guerra, T. Aida, and J. P. Freyer, 'Polarized angular dependent spectroscopy of epithelial cells and epithelial cell nuclei to determine the size scale of scattering structures,' J. Biomed. Opt. 7, 378-387 (2002).
[CrossRef] [PubMed]

J. Biophys. Biochem. Cytol. (1)

S. Inoue and W. L. Hyde, 'Studies on depolarization of light at microscope lens surfaces. 2. The simultaneous realization of high resolution and high sensitivity with the polarizing microscope,' J. Biophys. Biochem. Cytol. 3, 831-838 (1957).
[CrossRef] [PubMed]

J. Microsc. (2)

M. A. A. Neil, R. Juskaitis, M. J. Booth, T. Wilson, T. Tanaka, and S. Kawata, 'Adaptive aberration correction in a two-photon microscope,' J. Microsc. 200, 105-108 (2000).
[CrossRef] [PubMed]

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, 'Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,' J. Microsc. 206, 65-71 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (4)

R. V. Shack and B. C. Platt, 'Abstract. Production and use of a lenticular Hartmann screen,' J. Opt. Soc. Am. 61, 656-660 (1971).

R. Cubalchini, 'Modal wavefront estimation from phase derivative measurements,' J. Opt. Soc. Am. 69, 973-977 (1979).
[CrossRef]

R. J. Noll, 'Zernike polynomials and atmospheric-turbulence,' J. Opt. Soc. Am. 66, 207-211 (1976).
[CrossRef]

R. C. Jones, 'A new calculus for the treatment of optical systems I. Description and discussion of the calculus,' J. Opt. Soc. Am. 31, 488-493 (1941).
[CrossRef]

J. Opt. Soc. Am. A (5)

Math. Ann. (1)

A. Khintchine, 'Correlation theories of stationary stochastic processes,' Math. Ann. 109, 604-615 (1934).
[CrossRef]

Microsc. Res. Tech. (1)

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, 'Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,' Microsc. Res. Tech. 67, 36-44 (2005).
[CrossRef] [PubMed]

Nat. Neurosci. (1)

J. J. Zhu, J. A. Esteban, Y. Hayashi, and R. Malinow, 'Postnatal synaptic potentiation: Delivery of GluR4-containing AMPA receptors by spontaneous activity,' Nat. Neurosci. 3, 1098-1106 (2000).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

N. Shvartsman and I. Freund, 'Vortices in random wave-fields--nearest-neighbor anticorrelations,' Phys. Rev. Lett. 72, 1008-1011 (1994).
[CrossRef] [PubMed]

Proc. IEEE (1)

J. W. Hardy, 'Active optics: a new technology for the control of light (review article),' Proc. IEEE 66, 651-697 (1978).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (2)

M. J. Booth, M. A. A. Neil, R. Juskaitis, and T. Wilson, 'Adaptive aberration correction in a confocal microscope,' Proc. Natl. Acad. Sci. U.S.A. 99, 5788-5792 (2002).
[CrossRef] [PubMed]

M. Rueckel, J. A. Mack-Bucher, and W. Denk, 'Adaptive wavefront correction in two-photon microscopy using coherence-gated wavefront sensing,' Proc. Natl. Acad. Sci. U.S.A. 103, 17137-17142 (2006).
[CrossRef] [PubMed]

Proc. SPIE (2)

M. Rueckel and W. Denk, 'Coherence-gated wavefront sensing using a virtual Shack-Hartmann sensor,' Proc. SPIE 6306, 63060H (2006).
[CrossRef]

M. J. Booth, M. Schwertner, and T. Wilson, 'Specimen-induced aberrations and adaptive optics for microscopy,' Proc. SPIE 5894, 589403 (2005).
[CrossRef]

Rep. Prog. Phys. (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, 'Optical coherence tomography--principles and applications,' Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Science (2)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, 'Optical coherence tomography,' Science 254, 1178-1181 (1991).
[CrossRef]

W. Denk, J. H. Strickler, and W. W. Webb, 'Two-photon laser scanning fluorescence microscopy,' Science 248, 73-76 (1990).
[CrossRef] [PubMed]

Other (8)

M. Rueckel and W. Denk, 'Polarization effects in coherence-gated wave-front sensing,' in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings (CD-ROM), Technical Digest (Optical Society of America, 2005), paper AThC4.

E. Hecht, Optics (Addison-Wesley, 1990).

J. W. Goodman, Statistical Optics (Wiley, 2000).

D. Malacara, Optical Shop Testing (Wiley, 1992).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

C. F. Bohren and D. R. Huffmann, Absorption and Scattering by Small Particles (Wiley, 1983).

A. E. Siegman, Lasers (University Science, 1986).

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Figures (8)

Fig. 1
Fig. 1

(a) Calculation of τ ( k ) for sample light scattered by a scatterer at P ( x , y , z ) based on ray tracing. The incident ray passes the BFP at point A, is refracted by the objective lens toward the focus F, scatters at P in the direction w, and crosses the BFP at B on its return path (red online). For the CG centered at the focus the reference light takes a path that equals in length C-F-C (blue online) and A-F-B. Typical CVs, shown in cross section, for detection points (b) in the center and (c) at the edge of the BFP of the objective, respectively. The dotted curves correspond to γ ( τ ) = 0.5 and delineate the CV. The coherence length was 50 μ m (FWHM, normalized Gaussian self-coherence function), corresponding to a CG length of 18.8 μ m within the specimen, and the CG position was + 5 μ m .

Fig. 2
Fig. 2

(a), (c), (e) Number of phase singularities and (b), (d), (f) size of c 4 as they change with (a), (b) CG position, (c), (d), coherence length, (e), (f), and density of scatterers. Plotted are c 4 values as determined by estimating the centroid and those determined by fitting the peak by a Gaussian distribution. When not varied, the CG position, the density of scatterers, and the coherence length were + 5 μ m , 100 μ m 3 , and 30 μ m (FWHM), respectively. Linear fits to the data in (b) yield slopes of 119 ± 0.002 nm μ m (centroid) and of 78 ± 0.003 nm μ m (peak fit). Error bars are not shown in (b) because they are too small. (g) Experimentally measured c 4 as a function of the CG position for a scattering sample with 110 nm scattering beads (crosses, black) and for a chemically fixed organotypic rat hippocampus slice (dots, red). Also shown are linear fits to the data giving slopes of 101 ± 0.001 and 117 ± 0.001 nm μ m (very close to the theoretical expected value of 119 ± 0.002 nm μ m ). Note that the vertical displacement between the data series is not meaningful, since only relative CG positions were measured. Except for (b) and (g), the data points were connected by straight lines.

Fig. 3
Fig. 3

Asymmetric diffraction patterns for off-axis vSHS lenslets. (a) Illustration of how the CV is projected through a sublens. Peak fit (P) and centroid estimation (C) of the diffraction pattern differ. (b) Diffraction pattern numerically calculated for a edge lenslet and averaged over nine different scatterer ensembles for a CG position, density of scatterers, and coherence length (FWHM) of + 5 μ m , 100 μ m 3 , and 24 μ m , respectively. Note that, for better visibility, CV and focal cone are not drawn to scale.

Fig. 4
Fig. 4

MCSs showing the influence of distortions on the CGWS-measurement process. (a) The CV is deformed by distortions, and due to the extended CV distortions are averaged laterally, depending on the distance of the distortion layer to the focus. (b) The aberrations (Zernike coefficients c 5 to c 8 ) as a function of the coherence length with the phase plate located in the BFP. The dashed lines show the expected aberrations due to a single pass through a phase plate with c 6 = 0.5 μ m and c 8 = 0.3 μ m , respectively. The density of scatterers was 10 μ m 3 , and the CG was at + 5 μ m . (c) Phase plates with c 6 = 0.5 μ m and c 8 = 0.3 μ m located at varying distances above the focus and simulated for CG positions at 0 and 5 μ m . The coherence length was 50 μ m , and the density of scatterers was 1 μ m 3 . Only Zernike coefficients c 6 and c 8 are shown. The dashed lines indicate the expected aberrations for c 6 and c 8 .

Fig. 5
Fig. 5

(a) Distortions due to a tilted glass plate (10°, 145 μ m thick) inserted between the objective and the 100 nm bead sample. (b) Aberrations (Zernike coefficients c 4 to c 11 ) determined by MCS for a coherence length of 6 μ m and a density of scatterers of 1 μ m 3 for CG positions in steps of 2 μ m . (c) Experimentally measured aberrations for the tilted glass plate with the same parameters for CG positions in steps 1 μ m . The data points were connected by straight lines.

Fig. 6
Fig. 6

Detected wavefronts using linearly polarized light. Measured wavefronts using scattering samples with (a) 100 nm and (b) 1 μ m sized beads. The CG position was varied in steps of 2 and 3 μ m , respectively. The absolute CG position was not determined, only relative positions are depicted. For comparison, wavefronts determined by MCSs using (c) 100 nm and (d) 1 μ m sized beads are shown. A scatterer density of 1 μ m 3 and a coherence length of 2 μ m were used for the calculation. The CG position was varied from 8 to 8 μ m in steps of 4 μ m . Only spurious (varying with CG position) Zernike modes are shown: c 6 and c 12 . The slopes, obtained by linear fitting, are (a) 8 ± 3 , (b) 37 ± 3 , (c) 24 ± 4 , and (d) 50 ± 5 nm μ m for c 6 and (a) 1 ± 1 , (b) 1 ± 1 , (c) 2 ± 1 , and (d) 6 ± 1 nm μ m for c 12 .

Fig. 7
Fig. 7

Detected wavefronts using circularly polarized light. Experimentally measured wavefronts using scattering samples with (a) 100 nm (b) 1 μ m sized beads. The CG position was varied in steps of 1 μ m . For comparison, wavefronts determined by MCSs using (c) 100 nm and (d) 1 μ m sized beads. A scatterer density of 1 μ m 3 and a coherence length of 2 μ m were used. The CG position was varied from 8 to 8 μ m in steps of 4 μ m . Only spurious (varying with CG position) Zernike modes are shown: c 6 and c 12 . Linear fitting gave slopes of (a) 2 ± 2 , (b), 6 ± 3 , (c) 1 ± 3 , and (d) 2 ± 3 nm μ m for c 6 , and (a) 1 ± 1 , (b) 3 ± 1 , (c) 0 ± 3 , and (d) 1 ± 2 nm μ m for c 12 .

Fig. 8
Fig. 8

Propagation of the polarization through sample arm. The local bases are depicted as small arrows (red online). The fixed basis is shown at the focus. The scattering angle is ϑ.

Equations (31)

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I ( u , v ) = I R ( u , v ) + I S ( u , v ) + 2 Re { E R * ( u , v , t ) k E S ( k ) [ u , v , t + τ ( k ) ( u , v ) ] ¯ } ,
γ ( τ ) = E * ( t ) E ( t + τ ) ¯ E * ( t ) E ( t ) ¯ = γ ( τ ) exp ( i ω τ ) ,
I = I S + I R + 2 Re [ E R k ( p R * p S ( k ) ) E S ( k ) γ ( τ ( k ) ) exp ( i ω τ ( k ) ) ] .
τ [ r ( k ) , w ( u , v ) ] = n c [ sgn ( z ) r ( k ) + r ( k ) w ( u , v ) ] + l l 0 c ,
E CGWS ( u , v ) k W P ( k ) ( u , v ) E S ( k ) ( u , v ) γ [ τ ( k ) ( u , v ) ] exp [ i ω τ ( k ) ( u , v ) ] .
E CGWS ( r , q ) = A E CGWS ( u , v ) exp [ i 2 π λ f ( u r + v q ) ] d u d v = k E S ( k ) ( r , q ) ,
I ( r , q ) = E CGWS ( r , q ) 2 = k E S ( k ) ( r , q ) 2 + m n m [ E S ( m ) ( r , q ) ] * E S ( n ) ( r , q ) .
I ( r , q ) = A d s d t exp [ i 2 π λ f ( s r + t q ) ] A d u d v E CGWS ( u , v ) E CGWS * ( u + s , v + t ) .
E CGWS ( x 1 ) E CGWS * ( x 2 ) = k E S ( k ) ( x 1 ) E S ( k ) * ( x 2 ) + m n m E S ( m ) ( x 1 ) E S ( n ) * ( x 2 ) = k E S ( k ) ( x 1 ) E S ( k ) * ( x 2 ) + m n m E S ( m ) ( x 1 ) E S ( n ) * ( x 2 ) = N E S ( x 1 ) E S * ( x 2 ) + N ( N 1 ) E S ( x 1 ) E S * ( x 2 ) = N ( α β ) + N 2 β ,
τ aberr. ( r , w ) = τ ( r , w ) + τ illum. ( r ) + τ scatt. ( r , w ) ,
α = E S ( 0 ) E S * ( 0 ) = 1 V ( z 0 ) V ( z 0 ) d x d y d z E S 2 ( z , 0 ) γ [ τ ( z , 0 ) ] 2 ,
β = E S ( 0 ) E S * ( 0 ) = 1 V ( z 0 ) V ( z 0 ) d x d y d z E S ( z , 0 ) γ [ τ ( z , 0 ) ] exp [ i ω τ ( z , 0 ) ] 2 ,
W P ( x , y , z , u , v , p R , p I ) = W P ( x , y , z , u , v , p R , p I ) ,
W P ( x , y , z , u , v , p R , p I ) = W P ( x , y , z , u , v , p R , p I ) ,
e r ( 1 ) = e z ; e φ ( 1 ) = e y ; e ϑ ( 1 ) = e x .
e r ( 2 ) = e r ( 1 ) ; e φ ( 2 ) = e φ ( 3 ) ; e ϑ ( 2 ) = e φ ( 2 ) × e r ( 2 ) .
e r ( 3 ) = sgn ( z ) x 2 + y 2 + z 2 ( x e x + y e y + z e z ) ;
e φ ( 3 ) = e r ( 1 ) × e r ( 3 ) e r ( 1 ) × e r ( 3 ) ; e ϑ ( 3 ) = e φ ( 3 ) × e r ( 3 ) .
e r ( 4 ) = e r ( 3 ) ; e φ ( 4 ) = e r ( 4 ) × e r ( 5 ) e r ( 4 ) × e r ( 5 ) ; e ϑ ( 4 ) = e φ ( 4 ) × e r ( 4 ) .
e r ( 5 ) = w x e x + w y e y + w z e z ; e φ ( 5 ) = e φ ( 4 ) ; e ϑ ( 5 ) = e φ ( 5 ) × e r ( 5 ) .
e r ( 6 ) = e r ( 5 ) ; e φ ( 6 ) = e φ ( 7 ) ; e ϑ ( 6 ) = e φ ( 6 ) × e r ( 6 ) .
e r ( 7 ) = ( x z w z w x ) e x + ( y z w z w y ) e y + f back e z ,
e φ ( 7 ) = e r ( 5 ) × e r ( 7 ) e r ( 5 ) × e r ( 7 ) ; e ϑ ( 7 ) = e φ ( 7 ) × e r ( 7 ) .
( a 1 b 1 ) = ( 1 0 0 1 ) ( a ini b ini )
( a 2 b 2 ) = ( cos ( β ) sin ( β ) sin ( β ) cos ( β ) ) ( a 1 b 1 )
( a 3 b 3 ) = ( 1 0 0 1 ) ( a 2 b 2 ) .
( a 4 b 4 ) = ( cos ( φ ) sin ( φ ) sin ( φ ) cos ( φ ) ) ( a 3 b 3 )
( a 5 b 5 ) = ( S 2 ( ϑ ) 0 0 S 1 ( ϑ ) ) ( a 4 b 4 )
( a 6 b 6 ) = ( cos ( ε ) sin ( ε ) sin ( ε ) cos ( ε ) ) ( a 5 b 5 )
( a 7 b 7 ) = ( 1 0 0 1 ) ( a 6 b 6 ) .
( a f b f ) = ( e ϑ ( 7 ) e x e φ ( 7 ) e x e ϑ ( 7 ) e y e φ ( 7 ) e y ) ( a 7 b 7 )

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