Abstract

In line with the current interest in speckle motion in free-space propagation, it is proposed to investigate new aspects of the induced speckle motion due to rigid-body movements of a model using as a starting point the well-known concept of homology, first stated in holographic interferometry, the importance of which has only recently been realized in speckle metrology. At the outset, stating the homology conditions leads to a general expression relating the speckle shift to the geometrical parameters at the recording and to the six degrees of movement of the object surface. The expression presented in this explicit form is found to be true for both opaque and thin transparent diffusing models and for different conditions of illumination and observation. The study is then extended to (1) when the object is at rest, subjecting the illuminating source to a small displacement and (2) when the wavelength of the illuminating beam is changed between the two exposures. The case of the rigid-body displacement of the object is experimentally verified and the results are found to be in good agreement with theory. The viability of the method to make measurements is assessed, several applications are envisaged, and some advantages are pointed out.

© 1979 Optical Society of America

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References

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  1. K. A. Stetson, J. Opt. Soc. Am. 66, 1267 (1976).
    [CrossRef]
  2. D. A. Gregory, Opt. Laser Technol. 8, 201 (1976).
    [CrossRef]
  3. D. A. Gregory, Speckle Metrology, R. Erf, Ed. (Academic, New York, 1978), Chap. 8.
  4. K. A. Haines, B. P. Hildebrand, Appl. Opt. 5, 595 (1966).
    [CrossRef] [PubMed]
  5. K. A. Stetson, Optik 29, 386 (1969).
  6. C. Froehly, J. Monneret, J. Pasteur, J. C. Viénot, Opt. Acta 16, 343 (1969).
    [CrossRef]
  7. J. Tsujiuchi, N. Takeya, K. Matsuda, Opt. Acta 16, 709 (1969).
    [CrossRef]
  8. T. Tsuruta, N. Shiotake, Y. Itoh, Opt. Acta 16, 723 (1969).
    [CrossRef]
  9. J. Monneret, Opt. Commun. 2, 159 (1970).
    [CrossRef]
  10. K. A. Stetson, J. Opt. Soc. Am. 60, 1378 (1970).
    [CrossRef]
  11. S. Walles, Opt. Acta 17, 899 (1970).
    [CrossRef]
  12. M. Dubas, W. Schumann, Opt. Acta 22, 807 (1975).
    [CrossRef]
  13. K. A. Stetson, J. Opt. Soc. Am. 66, 626 (1976).
    [CrossRef]
  14. K. A. Stetson, J. Opt. Soc. Am. 67, 1587 (1977).
    [CrossRef]
  15. E. Archbold, E. Ennos, A. Virdee, Proc. Soc. Photo Opt. Instrum. Eng. 136, 258 (1977).
  16. M. Menu, M. L. Roblin, J. Opt. 10, 1 (1979).
    [CrossRef]
  17. G. Tribillon, Opt. Commun. 11, 172 (1974).
    [CrossRef]
  18. J. A. Méndez, M. L. Roblin, Opt. Commun. 13, 142 (1975).
    [CrossRef]
  19. D. Léger, E. Mathieu, J. C. Perrin, Appl. Opt. 14, 872 (1975).
    [CrossRef] [PubMed]
  20. D. Denby, G. E. Quintanilla, J. N. Butters, The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U. P., London, 1976).
  21. G. Tribillon, Proc. Soc. Photo Opt. Instrum. Eng. 136, 286 (1977).
  22. M. L. Roblin, G. Schalow, B. Chourabi, J. Opt. 8, 149 (1977).
    [CrossRef]
  23. A. K. Agarwal, P. C. Gupta, Opt. Commun. 17, 277 (1976).
    [CrossRef]
  24. See, for example, Ref. 7 or 8.
  25. M. Françon, Speckle (Masson, Paris, 1978).
  26. J. A. Méndez, M. L. Roblin, Opt. Commun. 15, 226 (1975).
    [CrossRef]
  27. K. A. Stetson, J. Opt. Soc. Am. 64, 857 (1974).
    [CrossRef]
  28. P. Jacquot, L. Pflug, P. K. Rastogi, Final Report submitted to Swiss National Foundation (November, 1978).

1979

M. Menu, M. L. Roblin, J. Opt. 10, 1 (1979).
[CrossRef]

1977

G. Tribillon, Proc. Soc. Photo Opt. Instrum. Eng. 136, 286 (1977).

M. L. Roblin, G. Schalow, B. Chourabi, J. Opt. 8, 149 (1977).
[CrossRef]

E. Archbold, E. Ennos, A. Virdee, Proc. Soc. Photo Opt. Instrum. Eng. 136, 258 (1977).

K. A. Stetson, J. Opt. Soc. Am. 67, 1587 (1977).
[CrossRef]

1976

K. A. Stetson, J. Opt. Soc. Am. 66, 626 (1976).
[CrossRef]

K. A. Stetson, J. Opt. Soc. Am. 66, 1267 (1976).
[CrossRef]

A. K. Agarwal, P. C. Gupta, Opt. Commun. 17, 277 (1976).
[CrossRef]

D. A. Gregory, Opt. Laser Technol. 8, 201 (1976).
[CrossRef]

1975

J. A. Méndez, M. L. Roblin, Opt. Commun. 15, 226 (1975).
[CrossRef]

M. Dubas, W. Schumann, Opt. Acta 22, 807 (1975).
[CrossRef]

D. Léger, E. Mathieu, J. C. Perrin, Appl. Opt. 14, 872 (1975).
[CrossRef] [PubMed]

J. A. Méndez, M. L. Roblin, Opt. Commun. 13, 142 (1975).
[CrossRef]

1974

1970

J. Monneret, Opt. Commun. 2, 159 (1970).
[CrossRef]

S. Walles, Opt. Acta 17, 899 (1970).
[CrossRef]

K. A. Stetson, J. Opt. Soc. Am. 60, 1378 (1970).
[CrossRef]

1969

K. A. Stetson, Optik 29, 386 (1969).

C. Froehly, J. Monneret, J. Pasteur, J. C. Viénot, Opt. Acta 16, 343 (1969).
[CrossRef]

J. Tsujiuchi, N. Takeya, K. Matsuda, Opt. Acta 16, 709 (1969).
[CrossRef]

T. Tsuruta, N. Shiotake, Y. Itoh, Opt. Acta 16, 723 (1969).
[CrossRef]

1966

Agarwal, A. K.

A. K. Agarwal, P. C. Gupta, Opt. Commun. 17, 277 (1976).
[CrossRef]

Archbold, E.

E. Archbold, E. Ennos, A. Virdee, Proc. Soc. Photo Opt. Instrum. Eng. 136, 258 (1977).

Butters, J. N.

D. Denby, G. E. Quintanilla, J. N. Butters, The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U. P., London, 1976).

Chourabi, B.

M. L. Roblin, G. Schalow, B. Chourabi, J. Opt. 8, 149 (1977).
[CrossRef]

Denby, D.

D. Denby, G. E. Quintanilla, J. N. Butters, The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U. P., London, 1976).

Dubas, M.

M. Dubas, W. Schumann, Opt. Acta 22, 807 (1975).
[CrossRef]

Ennos, E.

E. Archbold, E. Ennos, A. Virdee, Proc. Soc. Photo Opt. Instrum. Eng. 136, 258 (1977).

Françon, M.

M. Françon, Speckle (Masson, Paris, 1978).

Froehly, C.

C. Froehly, J. Monneret, J. Pasteur, J. C. Viénot, Opt. Acta 16, 343 (1969).
[CrossRef]

Gregory, D. A.

D. A. Gregory, Opt. Laser Technol. 8, 201 (1976).
[CrossRef]

D. A. Gregory, Speckle Metrology, R. Erf, Ed. (Academic, New York, 1978), Chap. 8.

Gupta, P. C.

A. K. Agarwal, P. C. Gupta, Opt. Commun. 17, 277 (1976).
[CrossRef]

Haines, K. A.

Hildebrand, B. P.

Itoh, Y.

T. Tsuruta, N. Shiotake, Y. Itoh, Opt. Acta 16, 723 (1969).
[CrossRef]

Jacquot, P.

P. Jacquot, L. Pflug, P. K. Rastogi, Final Report submitted to Swiss National Foundation (November, 1978).

Léger, D.

Mathieu, E.

Matsuda, K.

J. Tsujiuchi, N. Takeya, K. Matsuda, Opt. Acta 16, 709 (1969).
[CrossRef]

Méndez, J. A.

J. A. Méndez, M. L. Roblin, Opt. Commun. 13, 142 (1975).
[CrossRef]

J. A. Méndez, M. L. Roblin, Opt. Commun. 15, 226 (1975).
[CrossRef]

Menu, M.

M. Menu, M. L. Roblin, J. Opt. 10, 1 (1979).
[CrossRef]

Monneret, J.

J. Monneret, Opt. Commun. 2, 159 (1970).
[CrossRef]

C. Froehly, J. Monneret, J. Pasteur, J. C. Viénot, Opt. Acta 16, 343 (1969).
[CrossRef]

Pasteur, J.

C. Froehly, J. Monneret, J. Pasteur, J. C. Viénot, Opt. Acta 16, 343 (1969).
[CrossRef]

Perrin, J. C.

Pflug, L.

P. Jacquot, L. Pflug, P. K. Rastogi, Final Report submitted to Swiss National Foundation (November, 1978).

Quintanilla, G. E.

D. Denby, G. E. Quintanilla, J. N. Butters, The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U. P., London, 1976).

Rastogi, P. K.

P. Jacquot, L. Pflug, P. K. Rastogi, Final Report submitted to Swiss National Foundation (November, 1978).

Roblin, M. L.

M. Menu, M. L. Roblin, J. Opt. 10, 1 (1979).
[CrossRef]

M. L. Roblin, G. Schalow, B. Chourabi, J. Opt. 8, 149 (1977).
[CrossRef]

J. A. Méndez, M. L. Roblin, Opt. Commun. 15, 226 (1975).
[CrossRef]

J. A. Méndez, M. L. Roblin, Opt. Commun. 13, 142 (1975).
[CrossRef]

Schalow, G.

M. L. Roblin, G. Schalow, B. Chourabi, J. Opt. 8, 149 (1977).
[CrossRef]

Schumann, W.

M. Dubas, W. Schumann, Opt. Acta 22, 807 (1975).
[CrossRef]

Shiotake, N.

T. Tsuruta, N. Shiotake, Y. Itoh, Opt. Acta 16, 723 (1969).
[CrossRef]

Stetson, K. A.

Takeya, N.

J. Tsujiuchi, N. Takeya, K. Matsuda, Opt. Acta 16, 709 (1969).
[CrossRef]

Tribillon, G.

G. Tribillon, Proc. Soc. Photo Opt. Instrum. Eng. 136, 286 (1977).

G. Tribillon, Opt. Commun. 11, 172 (1974).
[CrossRef]

Tsujiuchi, J.

J. Tsujiuchi, N. Takeya, K. Matsuda, Opt. Acta 16, 709 (1969).
[CrossRef]

Tsuruta, T.

T. Tsuruta, N. Shiotake, Y. Itoh, Opt. Acta 16, 723 (1969).
[CrossRef]

Viénot, J. C.

C. Froehly, J. Monneret, J. Pasteur, J. C. Viénot, Opt. Acta 16, 343 (1969).
[CrossRef]

Virdee, A.

E. Archbold, E. Ennos, A. Virdee, Proc. Soc. Photo Opt. Instrum. Eng. 136, 258 (1977).

Walles, S.

S. Walles, Opt. Acta 17, 899 (1970).
[CrossRef]

Appl. Opt.

J. Opt.

M. L. Roblin, G. Schalow, B. Chourabi, J. Opt. 8, 149 (1977).
[CrossRef]

M. Menu, M. L. Roblin, J. Opt. 10, 1 (1979).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

C. Froehly, J. Monneret, J. Pasteur, J. C. Viénot, Opt. Acta 16, 343 (1969).
[CrossRef]

J. Tsujiuchi, N. Takeya, K. Matsuda, Opt. Acta 16, 709 (1969).
[CrossRef]

T. Tsuruta, N. Shiotake, Y. Itoh, Opt. Acta 16, 723 (1969).
[CrossRef]

S. Walles, Opt. Acta 17, 899 (1970).
[CrossRef]

M. Dubas, W. Schumann, Opt. Acta 22, 807 (1975).
[CrossRef]

Opt. Commun.

J. Monneret, Opt. Commun. 2, 159 (1970).
[CrossRef]

G. Tribillon, Opt. Commun. 11, 172 (1974).
[CrossRef]

J. A. Méndez, M. L. Roblin, Opt. Commun. 13, 142 (1975).
[CrossRef]

A. K. Agarwal, P. C. Gupta, Opt. Commun. 17, 277 (1976).
[CrossRef]

J. A. Méndez, M. L. Roblin, Opt. Commun. 15, 226 (1975).
[CrossRef]

Opt. Laser Technol.

D. A. Gregory, Opt. Laser Technol. 8, 201 (1976).
[CrossRef]

Optik

K. A. Stetson, Optik 29, 386 (1969).

Proc. Soc. Photo Opt. Instrum. Eng.

G. Tribillon, Proc. Soc. Photo Opt. Instrum. Eng. 136, 286 (1977).

E. Archbold, E. Ennos, A. Virdee, Proc. Soc. Photo Opt. Instrum. Eng. 136, 258 (1977).

Other

D. Denby, G. E. Quintanilla, J. N. Butters, The Engineering Uses of Coherent Optics, E. R. Robertson, Ed. (Cambridge U. P., London, 1976).

D. A. Gregory, Speckle Metrology, R. Erf, Ed. (Academic, New York, 1978), Chap. 8.

P. Jacquot, L. Pflug, P. K. Rastogi, Final Report submitted to Swiss National Foundation (November, 1978).

See, for example, Ref. 7 or 8.

M. Françon, Speckle (Masson, Paris, 1978).

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Figures (9)

Fig. 1
Fig. 1

Schematic of speckle recording in free-space propagation. The object undergoes a small rigid-body movement between the two exposures.

Fig. 2
Fig. 2

Fourier filtering setup for visualizing the correlation fringes.

Fig. 3
Fig. 3

Definition of the notations used. (a) The components of displacement; (b) coordinates of the illuminating source; (c) coordinates of the observation point So in the observation system [So,uvw].

Fig. 4
Fig. 4

Recording geometry of speckle related to two different states of the object: (a) in the initial state in the coordinate system [O,xyz]; (b) in the final state in the coordinate system [O′,XYZ].

Fig. 5
Fig. 5

Schematic of speckle recording for the object at rest and the illuminating source undergoing a small displacement between the two exposures.

Fig. 6
Fig. 6

Correlation fringes for each of the six components of displacement (T1,T2,T3,R1,R2,R3).

Fig. 7
Fig. 7

Verification of the parallelism of the diffracted waves in propagation in the case of T1 for a thin transparent diffusing model (collimated illumination and observation at zero incidence): (a) theoretical, - -; experimental, ×; (b) observed fringes.

Fig. 8
Fig. 8

(a) Sketch of the photographic plate showing the points of measurement; (b) correlation fringes representing the in-plane components of rotation R3 at these points in the case of an opaque diffusing object.

Fig. 9
Fig. 9

(a) Same as Fig. 8(a); (b) correlation fringes due to T3 in the case of a thin transparent diffusing object.

Tables (1)

Tables Icon

Table I Insensitivity Characteristics of the Method to Certain Displacement Components for Several Geometrical Configurations of Incident Wavefront and Observation Space

Equations (76)

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Δ u = λ f p cos γ ,             Δ v = λ f p sin λ ,
[ R ] = [ 1 R 3 - R 2 - R 3 1 R 1 R 2 - R 1 1 ] ;             [ R ] - 1 = [ 1 - R 3 R 2 R 3 1 - R 1 - R 2 R 1 1 ] ;             [ T ] = [ T 1 T 2 T 3 ] ,
[ X Y Z ] = [ R ] [ x y z ] ;             [ x y z ] = [ R ] - 1 [ X Y Z ] .
[ x y z ] = [ r ] [ u v w ] ;             [ u v w ] = [ r ] - 1 [ x y z ] ,
[ r ] = [ n o σ o - l o m o σ o l o 0 σ o m o - l o σ o - m o n o σ o n o ] ;             [ r ] - 1 = [ n o σ o 0 - l o σ o - l o m o σ o σ o - m o n o σ o l o m o n o ] ,
U ( S o ) U ( S o ) ,
S e P + P S o = S e P + P S o r e + r o = r e + r o }
ρ e + ρ o = ρ e + ρ o ,
l e + l o = l e + l o ,
m e + m o = m e + m o .
ρ e = ρ e - T · s ^ e = ρ e - ( T 1 l e + T 2 m e + T 3 n e ) ,
[ ρ e l e ρ e m e ρ e n e ] = [ R ] [ ρ e l e ρ e m e ρ e n e ] - [ T ] = [ ρ e l e + ρ e m e R 3 - ρ e n e R 2 - T 1 , ρ e m e - ρ e l e R 3 + ρ e n e R 1 - T 2 ρ e n e + ρ e l e R 2 - ρ e m e R 1 - T 3 ] .
l e = l e - n e R 2 + m e R 3 + [ ( l e 2 - 1 ) T 1 + l e m e T 2 + l e n e T 3 ] / ρ e ,
m e = m e + n e R 1 - l e R 3 + [ l e m e T 1 + ( m e 2 - 1 ) T 2 + m e n e T 3 ] / ρ e ,
n e = n e - m e R 1 + l e R 2 + [ l e n e T 1 + m e n e T 2 + ( n e 2 - 1 ) T 3 ] / ρ e .
l e 2 + m e 2 + n e 2 = 1.
l e = l e + e 1 ,
m e = m e + e 2 ,
n e = n e + e 3 ,
l e e 1 + m e e 2 + n e e 3 = 0.
l o = l o + o 1 ,
m o = m o + o 2 ,
n o = n o + o 3 .
l o o 1 + m o o 2 + n o o 3 = 0.
o 1 = - e 1 ,
o 2 = - e 2 .
o 3 = - [ ( l o o 1 + m o o 2 ) / n o ] .
l o = l o + n e R 2 - m e R 3 - [ ( l e 2 - 1 ) T 1 + l e m e T 2 + l e n e T 3 ] / ρ e ,
m o = m o - n e R 1 + l e R 3 - [ l e m e T 1 + ( m e 2 - 1 ) T 2 + m e n e T 3 ] / ρ e ,
n o = n o + n e m o n o R 1 - n e l o n o R 2 - ( l e m o - m e l o ) n o R 3 + [ ( l e 2 - 1 ) l o + l e m e m o ] T 1 + [ l e m e l o + ( m e 2 - 1 ) m o ] T 2 + [ l e n e l o + m e n e m o ] T 3 ρ e n o .
ρ o = ρ o + ( T 1 l e + T 2 m e + T 3 n e ) .
ρ R = O S R = ρ o - T · s ^ o = ρ o - ( l o T 1 + m o T 2 + n o T 3 ) .
Δ w = ρ o - ρ R = ( l e + l o ) T 1 + ( m e + m o ) T 2 + ( n e + n o ) T 3 = ( s ^ e + s ^ o ) · T
ρ C = ( 2 λ ρ o 2 ) / l 2 ,
O S R [ O , X Y Z ] | ρ o l o + ρ o ( n e R 2 - m e R 3 ) - [ ( l e 2 - 1 ) T 1 + l e m e T 2 + l e n e T 3 ] ρ o ρ e - l o ( l o T 1 + m o T 2 + n o T 3 ) ρ o m o + ρ o ( - n e R 1 + l e R 3 ) - [ l e m e T 1 + ( m e 2 - 1 ) T 2 + n e m e T 3 ] ρ o ρ e - m o ( l o T 1 + m o T 2 + n o T 3 ) ρ o n o + ρ o n o [ n e m o R 1 - n e l o R 2 - ( l e m o - l o m e ) R 3 ] + ρ o n o ρ e [ l o ( l e 2 - 1 ) + m o l e m e ] T 1 + ρ o n o ρ e [ l o l e m e + ( m e 2 - 1 ) m o ] T 2 + ρ a n o ρ e [ l o n e l e + n e m e m o ] T 3 - n o ( l o T 1 + m o T 2 + n o T 3 ) .
[ O S R ] = [ R ] - 1 [ O S R ] [ O , x y z ] [ O , X Y Z ] + T ,
[ Δ u Δ v 0 ] = [ S o S R ] [ S o , u v ] = [ r ] - 1 [ [ O S R ] [ O , x y z ] - [ ρ o l o ρ o m o ρ o n o ] ] .
[ Δ u Δ v Δ w ] = [ C ]             [ R 1 R 2 R 3 T 1 T 2 T 3 ]
[ Δ u Δ v Δ w ] = [ Δ u Δ v Δ w ] rot . + [ Δ u Δ v Δ w ] trans . ,
[ Δ u Δ v Δ w ] rot . = [ C ] rot . [ R 1 R 2 R 3 ] ;             [ Δ u Δ v Δ w ] trans . = [ C ] trans . [ T 1 T 2 T 3 ] ,
[ C ] rot . = ρ o σ o [ - m o l o ( 1 + n e n o ) σ o 2 ( 1 + n e n o ) l o n o ( m o l e - l o m e ) - n o ( m e + m o ) - ( n o + n e ) 0 l o + l e 0 0 0 ] ,
[ C ] trans . = ρ o ρ e σ o [ σ o 2 ( 1 - l e 2 ) + l o m o l e m e n o + n o ρ e ρ o l o m o ( 1 - m e 2 ) - σ o 2 l e m e n o - n e n o ( σ o 2 l e + l o m o m e ) - l o ρ e ρ o - l e m e - l o m o ρ e ρ o ( 1 - m e 2 ) + σ o 2 ρ e ρ o - n e m e - m o n o ρ e ρ o ( l o + l e ) ρ e σ o ρ a ( m o + m e ) ρ e σ o ρ o ( n o + n e ) ρ e σ o ρ o ] ,
[ C ] rot . = ρ o [ 0 1 + n e n o 0 - ( n o + n e ) 0 l o + l e 0 0 0 ] ,
[ C ] trans . = ρ 0 ρ e [ n e 2 n o + n o ρ e ρ o 0 - n e l e n o - l o ρ e ρ o 0 1 + ρ e ρ o 0 ( l o + l e ) ρ e ρ o 0 ( n o + n e ) ρ e ρ o ] .
[ S e [ S e , U V W ] | ρ e R H ρ e R V T L ,
S e [ O , x y z ] | ρ e l e + n e σ e ρ e R H - l e m e σ e ρ e R V + l e T L ρ e m e + σ e ρ e R V + m e T L ρ e n e - l e σ e ρ e R H - m e n e σ e ρ e R V + n e T L
e 1 = n e σ e R H - l e m e σ e R V + l e ρ e T L ,
e 2 = σ e R V + m e ρ e T L .
S o [ O , x y z ] | ρ o ( l o - e 1 ) ρ o ( m o - e 2 ) ρ o ( n o - e 3 ) ,
[ Δ u Δ v ] = [ - n o ρ o σ o e 1 + l o ρ o σ o e 3 l o m o ρ o σ o e 1 - σ o ρ o e 2 + m o n o ρ o σ o e 3 ] .
[ Δ u Δ v Δ w ] = [ C ] [ R H R V T L ] ,
[ C ] = ρ o [ - n e σ o n o σ e 1 n o ( l e m e σ o σ e - l o m o σ e σ o ) - 1 ρ e n o ( l e σ o + m o m e l o σ o ) 0 - σ e σ o - m e ρ e σ o 0 0 1 ] .
[ C ] = ρ o [ - n e n o 0 - l e ρ e n o 0 - 1 0 0 0 1 ] .
K 1 ( ρ e + ρ o ) = K 2 ( ρ e + ρ o ) ,
K 1 ( l e + l o ) = K 2 ( l e + l o ) ,
K 1 ( m e + m o ) = K 2 ( m e + m o ) ,
o 1 = ( l e + l o ) Δ λ λ ,
o 2 = ( m e + m o ) Δ λ λ ,
o 3 = - ( l o 2 + m o 2 + l o l e + m o m e ) n o Δ λ λ ,
O S R [ O , x y z ] | ρ o l o + ρ o o 1 ρ o m o + ρ o o 2 ρ o n o + ρ o o 3 ,
Δ u = [ l e + l o + m o ( l o m e - l e m o ) ] ρ o n ˙ o σ o Δ λ λ ,
Δ v = ( m e + m o ) ρ o σ o Δ λ λ ,
Δ w = ( ρ e + ρ o ) Δ λ λ .
Δ u = ( l e + l o ) ρ o n o Δ λ λ Δ v = 0 Δ w = ( ρ e + ρ o ) Δ λ λ } .
1 ρ e 1 ρ o 0.1 ,
Z max ρ e Z max ρ o 0.01 ,
Z max l 0.1.
[ C ] rot . = ρ o [ 0 - Δ R 0 Δ R 0 0 0 0 0 ] ,
[ C ] trans . = ρ o ρ e [ ( 1 - Δ T ) + ρ e ρ o 0 0 0 ( 1 - Δ T ) + ρ e ρ o 0 0 0 0 ] ,
Δ R = h ( n - 1 ) n ρ e ,
Δ T = h n ρ e ,
Δ w < ρ C .
Δ u > r C .
Δ u < ( Φ R ) / 5.
l o = u ρ o m o v ρ o n o = 1             Δ u = - m o ρ o R 3 Δ v = l o ρ o R 3 ;     [ u v ] becomes [ u - v R 3 v + u R 3 ]
l o = u ρ 0 m o = v ρ 0 n o = 1             Δ u = - l o T 3 Δ v = - m o T 3 ;     [ u v ] becomes [ u ( 1 - T 3 / ρ o ) v ( 1 - T 3 / ρ o ) ]

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