Abstract

The separation of multiple images is considered in the context of designing holographic image-focused particle diagnostic systems. Two techniques have been employed to determine the minimal directional cosine of the reference wave that ensures the separation of the first-order image of the particle field from the higher-order undesired images. The first approach is based on tracing the rays through the reconstructed images of the particles situated within the sample volume, whereas the second approach relies upon the analysis of the spectra of the reconstructed waves in the spatial-frequency domain. The results obtained by these techniques are compared and discussed.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. R. J. Adrian, "Particle imaging techniques for experimental fluid dynamics," Annu. Rev. Fluid Mech. 23, 261-304 (1991).
    [CrossRef]
  5. P. J. Santangelo and P. E. Sojka, "Holographic particle diagnostics," Prog. Energy Combust. Sci. 19, 587-603 (1993).
    [CrossRef]
  6. K. D. Hinsch, "Holographic particle image velocimetry," Meas. Sci. Technol. 13, 61-72 (2002).
    [CrossRef]
  7. W. L. Meng, H. Anderson, F. Hussain, and D. Liu, "Intrinsic speckle noise in in-line particle holography," J. Opt. Soc. Am. A 10, 2046-2058 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  10. J. D. Trolinger, B. L. Ravindra, D. McIntosh, and K. Witherow, "Holographic particle-image velocimetry in the first International Microgravity Laboratory aboard the Space Shuttle Discovery," Appl. Opt. 35, 681-689 (1996).
    [CrossRef] [PubMed]
  11. J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
    [CrossRef]
  12. Y. Pu and H. Meng, "An advanced off-axis holographic particle image velocimetry (HPIV) system," Exp. Fluids 29, 184-197 (2000).
    [CrossRef]
  13. K. Ellenrieder, J. Kostas, and J. Soria, "Measurements of a wall-bounded, turbulent, separated flow using HPIV," J. Turbul. 2, 1-15 (2001).
  14. R. Konrath, W. Schröder, and W. Limberg, "Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine," Exp. Fluids 33, 781-793 (2002).
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    [CrossRef] [PubMed]
  16. A. Svizher and J. Cohen, "Holographic particle image velocimetry system for measurements of hairpin vortices in air channel flow," Exp. Fluids 40, 708-722 (2006).
    [CrossRef]
  17. D. H. Barnhart, V. S. Chan, N. A. Halliwell, and J. M. Coupland, "Holographic velocimetry using object-conjugate reconstruction (OCR): a new approach for simultaneous, 3D displacement measurement in fluid and solid mechanics," Exp. Fluids 33, 770-780 (2002).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  22. E. B. Champagne, "Nonparaxial imaging, magnification, and aberration properties in holography," J. Opt. Soc. Am. 57, 51-55 (1967).
    [CrossRef]
  23. D. G. Falconer, "Role of the photographic process in holography," Photograph. Sci. Eng. 10, 133-139 (1966).
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    [CrossRef]
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2006 (1)

A. Svizher and J. Cohen, "Holographic particle image velocimetry system for measurements of hairpin vortices in air channel flow," Exp. Fluids 40, 708-722 (2006).
[CrossRef]

2003 (1)

2002 (3)

D. H. Barnhart, V. S. Chan, N. A. Halliwell, and J. M. Coupland, "Holographic velocimetry using object-conjugate reconstruction (OCR): a new approach for simultaneous, 3D displacement measurement in fluid and solid mechanics," Exp. Fluids 33, 770-780 (2002).

K. D. Hinsch, "Holographic particle image velocimetry," Meas. Sci. Technol. 13, 61-72 (2002).
[CrossRef]

R. Konrath, W. Schröder, and W. Limberg, "Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine," Exp. Fluids 33, 781-793 (2002).

2001 (1)

K. Ellenrieder, J. Kostas, and J. Soria, "Measurements of a wall-bounded, turbulent, separated flow using HPIV," J. Turbul. 2, 1-15 (2001).

2000 (2)

Y. Pu and H. Meng, "An advanced off-axis holographic particle image velocimetry (HPIV) system," Exp. Fluids 29, 184-197 (2000).
[CrossRef]

K. Sholes and P. V. Farrell, "Optical alignment-induced errors in holographic particle image velocimetry," Appl. Opt. 39, 5685-5693 (2000).
[CrossRef]

1998 (1)

C. S. Moraitis, "Multiple exposure aerodynamics particle holography," in Advanced Measurement Techniques, VKI RP 1998-06, F.Breugelmans, ed. (von Karman Institute for Fluid Dynamics, 1998).

1997 (1)

J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
[CrossRef]

1996 (2)

1994 (2)

1993 (3)

P. J. Santangelo and P. E. Sojka, "Holographic particle diagnostics," Prog. Energy Combust. Sci. 19, 587-603 (1993).
[CrossRef]

H. I. Bjelkhagen, Silver-Halide Recording Materials (Springer-Verlag, 1993), p. 67.

W. L. Meng, H. Anderson, F. Hussain, and D. Liu, "Intrinsic speckle noise in in-line particle holography," J. Opt. Soc. Am. A 10, 2046-2058 (1993).
[CrossRef]

1992 (1)

C. S. Vikram, Particle Field Holography, (Cambridge U. Press, 1992).
[CrossRef]

1991 (1)

R. J. Adrian, "Particle imaging techniques for experimental fluid dynamics," Annu. Rev. Fluid Mech. 23, 261-304 (1991).
[CrossRef]

1988 (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1988), pp. 402-403.

1969 (1)

1967 (2)

1966 (2)

1962 (1)

Adrian, R. J.

Anderson, H.

Barnhart, D. H.

D. H. Barnhart, V. S. Chan, N. A. Halliwell, and J. M. Coupland, "Holographic velocimetry using object-conjugate reconstruction (OCR): a new approach for simultaneous, 3D displacement measurement in fluid and solid mechanics," Exp. Fluids 33, 770-780 (2002).

D. H. Barnhart, R. J. Adrian, and G. C. Papen, "Phase-conjugate holographic system for high-resolution particle image velocimetry," Appl. Opt. 33, 7159-7169 (1994).
[CrossRef] [PubMed]

Bassini, A.

E. Masciadry, A. Bassini, S. Musazzi, E. Paganini, and U. Perini, "Recording of PIV images on photothermoplastic plates," Opt. Lasers Eng. 28, 497-501 (1996).
[CrossRef]

Belz, R. A.

Bjelkhagen, H. I.

H. I. Bjelkhagen, Silver-Halide Recording Materials (Springer-Verlag, 1993), p. 67.

Champagne, E. B.

Chan, V. S.

D. H. Barnhart, V. S. Chan, N. A. Halliwell, and J. M. Coupland, "Holographic velocimetry using object-conjugate reconstruction (OCR): a new approach for simultaneous, 3D displacement measurement in fluid and solid mechanics," Exp. Fluids 33, 770-780 (2002).

Cohen, J.

A. Svizher and J. Cohen, "Holographic particle image velocimetry system for measurements of hairpin vortices in air channel flow," Exp. Fluids 40, 708-722 (2006).
[CrossRef]

Coupland, J. M.

D. H. Barnhart, V. S. Chan, N. A. Halliwell, and J. M. Coupland, "Holographic velocimetry using object-conjugate reconstruction (OCR): a new approach for simultaneous, 3D displacement measurement in fluid and solid mechanics," Exp. Fluids 33, 770-780 (2002).

Ellenrieder, K.

K. Ellenrieder, J. Kostas, and J. Soria, "Measurements of a wall-bounded, turbulent, separated flow using HPIV," J. Turbul. 2, 1-15 (2001).

Falconer, D. G.

D. G. Falconer, "Role of the photographic process in holography," Photograph. Sci. Eng. 10, 133-139 (1966).

Farmer, W. M.

Farrell, P. V.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1988), pp. 402-403.

Halliwell, N. A.

D. H. Barnhart, V. S. Chan, N. A. Halliwell, and J. M. Coupland, "Holographic velocimetry using object-conjugate reconstruction (OCR): a new approach for simultaneous, 3D displacement measurement in fluid and solid mechanics," Exp. Fluids 33, 770-780 (2002).

Hinsch, K. D.

K. D. Hinsch, "Holographic particle image velocimetry," Meas. Sci. Technol. 13, 61-72 (2002).
[CrossRef]

Hussain, F.

Katz, J.

J. Sheng, E. Malkiel, and J. Katz, "Single beam two-views holographic particle image velocimetry," Appl. Opt. 42, 235-250 (2003).
[CrossRef] [PubMed]

J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
[CrossRef]

Konrath, R.

R. Konrath, W. Schröder, and W. Limberg, "Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine," Exp. Fluids 33, 781-793 (2002).

Kostas, J.

K. Ellenrieder, J. Kostas, and J. Soria, "Measurements of a wall-bounded, turbulent, separated flow using HPIV," J. Turbul. 2, 1-15 (2001).

Leith, E. N.

Limberg, W.

R. Konrath, W. Schröder, and W. Limberg, "Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine," Exp. Fluids 33, 781-793 (2002).

Liu, D.

Malkiel, E.

Masciadry, E.

E. Masciadry, A. Bassini, S. Musazzi, E. Paganini, and U. Perini, "Recording of PIV images on photothermoplastic plates," Opt. Lasers Eng. 28, 497-501 (1996).
[CrossRef]

McIntosh, D.

Meng, H.

Y. Pu and H. Meng, "An advanced off-axis holographic particle image velocimetry (HPIV) system," Exp. Fluids 29, 184-197 (2000).
[CrossRef]

Meng, W. L.

Moraitis, C. S.

C. S. Moraitis, "Multiple exposure aerodynamics particle holography," in Advanced Measurement Techniques, VKI RP 1998-06, F.Breugelmans, ed. (von Karman Institute for Fluid Dynamics, 1998).

Musazzi, S.

E. Masciadry, A. Bassini, S. Musazzi, E. Paganini, and U. Perini, "Recording of PIV images on photothermoplastic plates," Opt. Lasers Eng. 28, 497-501 (1996).
[CrossRef]

Neumann, D. B.

Paganini, E.

E. Masciadry, A. Bassini, S. Musazzi, E. Paganini, and U. Perini, "Recording of PIV images on photothermoplastic plates," Opt. Lasers Eng. 28, 497-501 (1996).
[CrossRef]

Papen, G. C.

Perini, U.

E. Masciadry, A. Bassini, S. Musazzi, E. Paganini, and U. Perini, "Recording of PIV images on photothermoplastic plates," Opt. Lasers Eng. 28, 497-501 (1996).
[CrossRef]

Pu, Y.

Y. Pu and H. Meng, "An advanced off-axis holographic particle image velocimetry (HPIV) system," Exp. Fluids 29, 184-197 (2000).
[CrossRef]

Ravindra, B. L.

Santangelo, P. J.

P. J. Santangelo and P. E. Sojka, "Focused-image holography as a dense-spray diagnostic," Appl. Opt. 33, 4132-4136 (1994).
[CrossRef] [PubMed]

P. J. Santangelo and P. E. Sojka, "Holographic particle diagnostics," Prog. Energy Combust. Sci. 19, 587-603 (1993).
[CrossRef]

Schröder, W.

R. Konrath, W. Schröder, and W. Limberg, "Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine," Exp. Fluids 33, 781-793 (2002).

Sheng, J.

Sholes, K.

Sojka, P. E.

P. J. Santangelo and P. E. Sojka, "Focused-image holography as a dense-spray diagnostic," Appl. Opt. 33, 4132-4136 (1994).
[CrossRef] [PubMed]

P. J. Santangelo and P. E. Sojka, "Holographic particle diagnostics," Prog. Energy Combust. Sci. 19, 587-603 (1993).
[CrossRef]

Soria, J.

K. Ellenrieder, J. Kostas, and J. Soria, "Measurements of a wall-bounded, turbulent, separated flow using HPIV," J. Turbul. 2, 1-15 (2001).

Svizher, A.

A. Svizher and J. Cohen, "Holographic particle image velocimetry system for measurements of hairpin vortices in air channel flow," Exp. Fluids 40, 708-722 (2006).
[CrossRef]

Tao, B.

J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
[CrossRef]

Thompson, B. J.

Trolinger, J. D.

Upatnieks, J.

Vikram, C. S.

C. S. Vikram, Particle Field Holography, (Cambridge U. Press, 1992).
[CrossRef]

Ward, J. H.

Witherow, K.

Zhang, J.

J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
[CrossRef]

Zinky, W. R.

Annu. Rev. Fluid Mech. (1)

R. J. Adrian, "Particle imaging techniques for experimental fluid dynamics," Annu. Rev. Fluid Mech. 23, 261-304 (1991).
[CrossRef]

Appl. Opt. (7)

Exp. Fluids (5)

A. Svizher and J. Cohen, "Holographic particle image velocimetry system for measurements of hairpin vortices in air channel flow," Exp. Fluids 40, 708-722 (2006).
[CrossRef]

D. H. Barnhart, V. S. Chan, N. A. Halliwell, and J. M. Coupland, "Holographic velocimetry using object-conjugate reconstruction (OCR): a new approach for simultaneous, 3D displacement measurement in fluid and solid mechanics," Exp. Fluids 33, 770-780 (2002).

J. Zhang, B. Tao, and J. Katz, "Turbulent flow measurement in a square duct with hybrid holographic PIV," Exp. Fluids 23, 373-381 (1997).
[CrossRef]

Y. Pu and H. Meng, "An advanced off-axis holographic particle image velocimetry (HPIV) system," Exp. Fluids 29, 184-197 (2000).
[CrossRef]

R. Konrath, W. Schröder, and W. Limberg, "Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine," Exp. Fluids 33, 781-793 (2002).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

J. Turbul. (1)

K. Ellenrieder, J. Kostas, and J. Soria, "Measurements of a wall-bounded, turbulent, separated flow using HPIV," J. Turbul. 2, 1-15 (2001).

Meas. Sci. Technol. (1)

K. D. Hinsch, "Holographic particle image velocimetry," Meas. Sci. Technol. 13, 61-72 (2002).
[CrossRef]

Opt. Lasers Eng. (1)

E. Masciadry, A. Bassini, S. Musazzi, E. Paganini, and U. Perini, "Recording of PIV images on photothermoplastic plates," Opt. Lasers Eng. 28, 497-501 (1996).
[CrossRef]

Photograph. Sci. Eng. (1)

D. G. Falconer, "Role of the photographic process in holography," Photograph. Sci. Eng. 10, 133-139 (1966).

Prog. Energy Combust. Sci. (1)

P. J. Santangelo and P. E. Sojka, "Holographic particle diagnostics," Prog. Energy Combust. Sci. 19, 587-603 (1993).
[CrossRef]

Other (4)

C. S. Vikram, Particle Field Holography, (Cambridge U. Press, 1992).
[CrossRef]

C. S. Moraitis, "Multiple exposure aerodynamics particle holography," in Advanced Measurement Techniques, VKI RP 1998-06, F.Breugelmans, ed. (von Karman Institute for Fluid Dynamics, 1998).

H. I. Bjelkhagen, Silver-Halide Recording Materials (Springer-Verlag, 1993), p. 67.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1988), pp. 402-403.

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Figures (12)

Fig. 1
Fig. 1

Individual object waves. (a) Object positioning. Case 1: the object is positioned in front of the photoplate at point P O 1 . Case 2: the object is situated behind the photoplate at point P O 2 . (b) A perspective view of the spherical converging object wave emitting from point P O 1 ( x O 1 , y O 1 , z O 1 ) . B is the center of the individual hologram aperture associated with P O 1 . X , Y , Z is the local coordinate system assigned to this object point, having its origin at B. The photoplate is positioned in the X , Y plane.

Fig. 2
Fig. 2

Recording and phase-conjugate reconstruction of a single particle. The particle image position during the recording stage is denoted by P O . During the reconstruction stage real images of various diffraction orders are generated at points P 1 , P 2 , P 3 , P 4 , P 5 , and P 6 , respectively. Point P 1 marks the position of the virtual image corresponding to the reconstructed wave of order 1 . D h denotes the aperture of the individual hologram of the particle.

Fig. 3
Fig. 3

Recording and phase-conjugate reconstruction of the sample volume. Reconstructed waves corresponding only to the first and second diffraction orders are shown.

Fig. 4
Fig. 4

Tracing the rays through the object point P O (recording) and through its associated first and n th -order images denoted by P 1 and P n , respectively (reconstruction).

Fig. 5
Fig. 5

Criterion for image separation (Cr) versus the directional cosine of the reference wave ( α ) ; y O z O = 0 , S z O = 1.05 , D l S = 0.175 . (a) The effect of the image order n; x O z O = 0.05 . (b) The effect of the dimensionless parameter x O z O ; n = 2 .

Fig. 6
Fig. 6

Spectra of an individual phase hologram in the case where α > α marg max .

Fig. 7
Fig. 7

α cr f versus x O z O . (a) D l z O = ( D l S ) ( S z O ) = 0.184 < 0.466 corresponding to the plane z = z O min (see Fig. 3 and Table 1). The point indicated by the letter D corresponds to the upper left corner of the sample volume. (b) D l z O = 0.525 > 0.466 .

Fig. 8
Fig. 8

Two cases of recording and phase-conjugate reconstruction of the particle image formed by imaging optics having different diameters but the same NA.

Fig. 9
Fig. 9

Reduction percentage of α cr as a function of S z O calculated for three sets of the computational parameters. Point D on curve 2 corresponds to the upper left corner of the sample volume of the designed HPIV system. Computational parameters: y O z O = 0 ; curve 1, D l z O = 0.5 , x O z O = 0.05 ; curve 2, D l z O = 0.184 , x O z O = 0.05 ; curve 3, D l z O = 0.184 , x O z O = 0.15 .

Fig. 10
Fig. 10

Position of point G corresponding to the highest value of the left-hand side of inequality (B2). P O proj denotes the projection of object point P O on the hologram plane, and B marks the center of the individual hologram aperture associated with P O .

Fig. 11
Fig. 11

An example illustrating the behavior of α marg min and α marg max as functions of x O z O . D l z O = D l z O 1 = const.

Fig. 12
Fig. 12

Intervals of D l z O (within the image of the sample volume) associated with various values of x O z O = const .

Tables (1)

Tables Icon

Table 1 Range of Parameters for Calculating Cr

Equations (88)

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T a = n = + ( j ) n J n ( 2 v t e A R A O ) exp [ j n ( ϕ O ϕ R ) ] ,
T a A C exp [ j ϕ C ] = A C n = + ( j ) n J n ( 2 v t e A R A O ) exp [ j n ( ϕ O ϕ R Φ C n ) ] n = + A I n exp [ j ϕ I n ] .
A O exp [ j ϕ O ] = A O exp [ sign ( z O ) j k r obj ] ,
ϕ O = sign ( z O ) k [ 1 2 r O ( x h 2 + y h 2 ) x O r O x h y O r O y h ( x h 2 + y h 2 ) 2 4 ( x h 2 + y h 2 ) ( x O x h + y O y h ) + 4 ( x O x h + y O y h ) 2 8 r O 3 + ] .
ϕ I n = sign ( z O n ) k { n 2 r O ( x h 2 + y h 2 ) n [ x O r O sign ( z O ) ( α R + α C n ) ] x h n [ y O r O sign ( z O ) ( β R + β C n ) ] y h ( x h 2 + y h 2 ) 2 4 ( x h 2 + y h 2 ) ( x O x h + y O y h ) + 4 ( x O x h + y O y h ) 2 8 r O 3 } .
ϕ sph = sign ( z I n ) k [ 1 2 r I n ( x h 2 + y h 2 ) x I n r I n x h y I n r I n y h ( x h 2 + y h 2 ) 2 4 ( x h 2 + y h 2 ) ( x I n x h + y I n y h ) + 4 ( x I n x h + y I n y h ) 2 8 r I n 3 ] ,
r I n = r O n ,
x I n = x O sign ( z O ) r O ( α R + α C n ) ,
y I n = y O sign ( z O ) r O ( β R + β C n ) ,
sign ( z I n ) = sign ( z O n γ C ) .
( x I n r I n ) 2 + ( y I n r I n ) 2 < 1 .
[ x O r O sign ( z O ) ( α R + α C n ) ] 2 + [ y O r O sign ( z O ) ( β R + β C n ) ] 2 < 1 n 2 .
Δ ϕ = k [ r h 4 8 Sp + r h 3 2 ( C x cos φ + C y sin φ ) r h 2 2 ( A x cos 2 φ + A y sin 2 φ + A x y 2 cos φ sin φ ) ] ,
Sp = n r O 3 ( 1 n 2 ) ,
C x = n r O 2 [ x O r O ( 1 n 2 ) n 2 sign ( z O ) ( α R + α C n ) ] ,
C y = n r O 2 [ y O r O ( 1 n 2 ) n 2 sign ( z O ) ( β R + β C n ) ] ,
A x = n r O [ ( x O r O ) 2 ( 1 n 2 ) + 2 n 2 sign ( z O ) x O r O ( α R + α C n ) n 2 ( α R + α C n ) 2 ] ,
A y = n r O [ ( y O r O ) 2 ( 1 n 2 ) + 2 n 2 sign ( z O ) y O r O ( β R + β C n ) n 2 ( β R + β C n ) 2 ] ,
A x y = n r O { x O y O r O 2 ( 1 n 2 ) + n sign ( z O ) [ x O r O ( β R + β C n ) + y O r O ( α R + α C n ) ] n 2 ( α R + α C n ) ( β R + β C n ) } .
α R = α C = α ,
β R = β C = 0 ,
γ C = ( 1 α C 2 ) 1 2 < 0 .
r I n = r O n ,
x I n = x O sign ( z O ) r O α n 1 n ,
y I n = y O ,
sign ( z I n ) = sign ( z O n ) ,
[ x O r O sign ( z O ) α n 1 n ] 2 + [ y O r O ] 2 < 1 n 2 .
x = x + x O z O S ,
y = y + y O z O S ,
z = z + S .
Cr = 1 S ( A C n D l + D n 2 ) 0 ,
Cr = [ ( x I n z I n x O z O ) 2 + ( y I n z I n y O z O ) 2 ] 1 2 1 2 D 1 S ( 1 S z O 1 + S z O z O z I n + 1 ) ,
F n min = n λ ( α α marg max ) ,
F n max = n λ ( α α marg min ) for n > 0 ,
F n min = n λ ( α α marg min ) ,
F n max = n λ ( α α marg max ) for n < 0 ,
α > α cr f = n α marg max α marg min n 1 , for n > 1 ;
α > α cr f = α marg max , for n < 0 .
α α cr f = 2 α marg max α marg min = α marg max + λ B O ,
α cr f = 2 ( x O z O + 1 2 D l z O ) [ 1 + ( x O z O + 1 2 D l z O ) 2 ] 1 2 x O z O 1 2 D l z O [ 1 + ( x O z O 1 2 D l z O ) 2 ] 1 2 .
α α marg min < λ F medium max ,
α α marg max > λ F medium min ,
α > λ F medium min + ( α marg max ) D ,
α < λ F medium max ( α marg max ) D ,
α > ( 2 α marg max ) D ( α marg min ) D = λ ( B O ) D + ( α marg max ) D .
F medium max F medium min > 2 λ ( α marg max ) D , for F medium min > ( B O ) D ;
F medium max > ( B O ) D + 2 λ ( α marg max ) D , for F medium min < ( B O ) D .
lim S z O 1 α cr = α cr f .
x x I n = y y I n = z z I n ,
x C n = x I n z I n S ,
y C n = y I n z I n S ,
z C n = S .
x C n = ( x O z O x I n z I n ) S ,
y C n = ( y O z O y I n z I n ) S ,
z C n = 0 .
A C n = S [ ( x I n z I n x O z O ) 2 + ( y I n z I n y O z O ) 2 ] 1 2 .
r O z O = [ 1 + ( x O z O ) 2 + ( y O z O ) 2 ] 1 2 = [ x O 2 + y O 2 + z O 2 z O 2 ( 1 S z O ) 2 ] 1 2 = r O z O 1 S z O ,
x I n z O = ( 1 S z O ) ( x O z O α n 1 n r O z O ) ,
y I n z O = ( 1 S z O ) y O z O ,
z I n z O = sign ( z O n ) [ 1 n 2 ( r O z O ) 2 ( x I n z O ) 2 ( y I n z O ) 2 ] 1 2 = sign ( n ) ( 1 S z O ) [ 1 n 2 ( r O z O ) 2 ( x O z O α n 1 n r O z O ) 2 ( y O z O ) 2 ] 1 2 ,
x I n z I n = sign ( n ) x O z O α n 1 n r O z O [ 1 n 2 ( r O z O ) 2 ( x O z O α n 1 n r O z O ) 2 ( y O z O ) 2 ] 1 2 ,
y I n z I n = sign ( n ) y O z O [ 1 n 2 ( r O z O ) 2 ( x O z O α n 1 n r O z O ) 2 ( y O z O ) 2 ] 1 2 ,
z I n z O = sign ( n ) ( 1 S z O ) [ 1 n 2 ( r O z O ) 2 ( x O z O α n 1 n r O z O ) 2 ( y O z O ) 2 ] 1 2 ,
r O z O = [ 1 + ( x O z O ) 2 + ( y O z O ) 2 ] 1 2 .
( x O z O α n 1 n r O z O ) 2 + ( y O z O ) 2 < 1 n 2 ( r O z O ) 2 .
D h = S z O z O D l .
D n = S + z I n z I n D h .
D n = D l 1 S z O 1 + S z O z O z I n .
α marg max = sign ( x O + D l 2 ) cos [ cot 1 ( x O + D l 2 z O ) ] ,
α marg min = sign ( x O D l 2 ) cos [ cot 1 ( x O D l 2 z O ) ] .
cot 1 ( a ) = cos 1 [ a ( 1 + a 2 ) 1 2 ]
α marg max = ( x O z O + 1 2 D l z O ) [ 1 + ( x O z O + 1 2 D l z O ) 2 ] 1 2 ,
α marg min = x O z O 1 2 D l z O [ 1 + ( x O z O 1 2 D l z O ) 2 ] 1 2 .
x h 2 + y h 2 2 x O x h 2 y O y h < r O 2 .
P O proj G 2 2 ( x O 2 + y O 2 ) < z O 2 ,
P O proj G max = B G + P O proj B = D h 2 + ( x O 2 + y O 2 ) 1 2 .
D h 2 2 [ D h 2 ( x O 2 + y O 2 ) 1 2 ] 2 < z O 2 .
D l 2 2 { D l 2 [ ( x O ) 2 + ( y O ) 2 ] 1 2 } 2 ( z O ) 2 < 1 .
F n min = n λ ( α α marg max ) < 0 ,
F n max = n λ ( α α marg min ) > 0 .
α marg min D l z O = 1 2 [ 1 + ( x O z O 1 2 D l z O ) 2 ] 3 2 ,
α marg min x O z O = 1 [ 1 + ( x O z O 1 2 D l z O ) 2 ] 3 2 ,
α marg max D l z O = 1 2 [ 1 + ( x O z O + 1 2 D l z O ) 2 ] 3 2 ,
α marg max x O z O = 1 [ 1 + ( x O z O + 1 2 D l z O ) 2 ] 3 2
α cr f D l z O = 2 [ 1 + ( x O z O 1 2 D l z O ) 2 ] 3 2 + [ 1 + ( x O z O + 1 2 D l z O ) 2 ] 3 2 2 [ 1 + ( x O z O + 1 2 D l z O ) 2 ] 3 2 [ 1 + ( x O z O 1 2 D l z O ) 2 ] 3 2 ,
α cr f x O z O = 2 [ 1 + ( x O z O 1 2 D l z O ) 2 ] 3 2 [ 1 + ( x O z O + 1 2 D l z O ) 2 ] 3 2 [ 1 + ( x O z O + 1 2 D l z O ) 2 ] 3 2 [ 1 + ( x O z O 1 2 D l z O ) 2 ] 3 2 .
( x O z O ) max ( α cr f ) = 2 2 3 1 2 ( 2 2 3 + 1 ) D l z O [ 2 2 3 ( 2 2 3 1 ) 2 ( D l z O ) 2 1 ] 1 2 2.2 D l z O [ 4.6 ( D l z O ) 2 1 ] 1 2 ,
( x O z O ) min ( α cr f ) = 2 2 3 1 2 ( 2 2 3 + 1 ) D l z O + [ 2 2 3 ( 2 2 3 1 ) 2 ( D l z O ) 2 1 ] 1 2 2.2 D l z O + [ 4.6 ( D l z O ) 2 1 ] 1 2 .

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