Abstract

The phase degradation of an optical wave front distorted by turbulence in the propagation medium may be corrected in a piecewise-linear fashion by using an array of small circular mirrors. An option in the correction scheme is to compensate for overall tilt separately. We have evaluated power spectra and variances of the piston and tilt motions of the mirror segments as well as the motion of the overall tilt corrector. The form of the spectra for any propagation medium is an aperture integral of the product of the phase-difference power spectrum, describing the medium, and a generalized transfer function, representing the aperture and its segments. In the case of low-power atmospheric propagation, the necessary propagation results are linear in turbulence strength; hence the path may be sectioned into a large number of thin slices. A set of standard curves is found to represent a generalized slice, and the differential contributions may be summed to represent any propagation path. The standard curves, further modeled in terms of power-law dependencies, are practical for use on a desk calculator.

© 1976 Optical Society of America

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References

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  1. R. A. Muller and A. Buffington, J. Opt. Soc. Am. 64, 1200 (1974).
    [Crossref]
  2. J. W. Hardy, J. Feinleib, and J. C. Wyant, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB1.
  3. L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB2.
  4. J. E. Pearson, W. B. Bridges, L. S. Horowitz, T. J. Walsh, and R. F. Ogrodnik, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB5.
  5. D. L. Fried, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper TuA6.
  6. V. N. Mahajan, J. Opt. Soc. Am. 65, 271 (1975).
    [Crossref]
  7. D. L. Fried and H. T. Yura, J. Opt. Soc. Am. 62, 600 (1972).
    [Crossref]
  8. H. V. Hance and D. L. Fried, J. Opt. Soc. Am. 63, 1015 (1973).
    [Crossref]
  9. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (U. S. Dept. of Commerce, Washington, D. C., NTIS T68-50464, 1971).
  10. J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
    [Crossref]
  11. J. L. Bufton, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper WA3.
  12. R. S. Lawrence, G. R. Ochs, and S. F. Clifford, J. Opt. Soc. Am. 60, 826 (1970).
    [Crossref]
  13. R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
    [Crossref]
  14. S. F. Clifford, J. Opt. Soc. Am. 61, 1285 (1971).
    [Crossref]
  15. G. W. Reinhardt and S. A. Collins, J. Opt. Soc. Am. 62, 1526 (1972).
    [Crossref]
  16. R. F. Lutomirski and R. G. Buser, Appl. Opt. 13, 2869 (1974).
    [Crossref] [PubMed]
  17. R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
    [Crossref]
  18. D. L. Fried, Proc. IEEE 55, 57 (1967).
    [Crossref]
  19. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
    [Crossref]

1975 (1)

1974 (2)

1973 (1)

1972 (2)

1971 (1)

1970 (2)

1969 (1)

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[Crossref]

1968 (1)

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

1967 (2)

Bridges, W. B.

J. E. Pearson, W. B. Bridges, L. S. Horowitz, T. J. Walsh, and R. F. Ogrodnik, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB5.

Brown, W. P.

L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB2.

Buffington, A.

Bufton, J. L.

J. L. Bufton, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper WA3.

Buser, R. G.

Clifford, S. F.

Collins, S. A.

Feinleib, J.

J. W. Hardy, J. Feinleib, and J. C. Wyant, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB1.

Fried, D. L.

H. V. Hance and D. L. Fried, J. Opt. Soc. Am. 63, 1015 (1973).
[Crossref]

D. L. Fried and H. T. Yura, J. Opt. Soc. Am. 62, 600 (1972).
[Crossref]

D. L. Fried, Proc. IEEE 55, 57 (1967).
[Crossref]

D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
[Crossref]

D. L. Fried, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper TuA6.

Hance, H. V.

Hardy, J. W.

J. W. Hardy, J. Feinleib, and J. C. Wyant, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB1.

Harp, J. C.

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[Crossref]

Horowitz, L. S.

J. E. Pearson, W. B. Bridges, L. S. Horowitz, T. J. Walsh, and R. F. Ogrodnik, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB5.

Jenney, J. A.

L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB2.

Lawrence, R. S.

Lee, R. W.

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[Crossref]

Lutomirski, R. F.

Mahajan, V. N.

Miller, L.

L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB2.

Muller, R. A.

O’Meara, T. R.

L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB2.

Ochs, G. R.

Ogrodnik, R. F.

J. E. Pearson, W. B. Bridges, L. S. Horowitz, T. J. Walsh, and R. F. Ogrodnik, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB5.

Pearson, J. E.

J. E. Pearson, W. B. Bridges, L. S. Horowitz, T. J. Walsh, and R. F. Ogrodnik, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB5.

Reinhardt, G. W.

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[Crossref]

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (U. S. Dept. of Commerce, Washington, D. C., NTIS T68-50464, 1971).

Walsh, T. J.

J. E. Pearson, W. B. Bridges, L. S. Horowitz, T. J. Walsh, and R. F. Ogrodnik, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB5.

Wyant, J. C.

J. W. Hardy, J. Feinleib, and J. C. Wyant, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB1.

Yura, H. T.

Appl. Opt. (1)

J. Opt. Soc. Am. (8)

Proc. IEEE (4)

R. W. Lee and J. C. Harp, Proc. IEEE 57, 375 (1969).
[Crossref]

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

R. S. Lawrence and J. W. Strohbehn, Proc. IEEE 58, 1523 (1970).
[Crossref]

D. L. Fried, Proc. IEEE 55, 57 (1967).
[Crossref]

Other (6)

J. L. Bufton, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper WA3.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (U. S. Dept. of Commerce, Washington, D. C., NTIS T68-50464, 1971).

J. W. Hardy, J. Feinleib, and J. C. Wyant, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB1.

L. Miller, W. P. Brown, J. A. Jenney, and T. R. O’Meara, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB2.

J. E. Pearson, W. B. Bridges, L. S. Horowitz, T. J. Walsh, and R. F. Ogrodnik, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper ThB5.

D. L. Fried, Digest, Topical Meeting on Optical Propagation Through Turbulence, O. S. A., July1974, paper TuA6.

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Figures (13)

FIG. 1
FIG. 1

A one-dimensional representation of wave-front correction (vertical scale greatly expanded). (A) Initial wave front at aperture plane, (B) wave front after overall tilt correction; local piston and tilt shown for segment of size d = D/7, (C) residual wave front after compensation over all segments.

FIG. 2
FIG. 2

Drive distances of piston and single-point phase versus location on aperture for d = D/10 and d = 0. Symbols: + piston, overall tilt in; (– · – · –) single-point phase, overall tilt in; × piston, overall tilt removed; (– – –) single-point phase, overall tilt removed. Overall tilt representation (—) σφ = k σα(D) |r1|.

FIG. 3
FIG. 3

Drive distance of tilt versus location on aperture for d = D/10. Symbols: (– · –· –) tilt in either orientation for overall tilt left in; + azimuthal tilt for overall tilt removed; × radial tilt for overall tilt removed; (—) overall tilt representation σα(d) = σα(D).

FIG. 4
FIG. 4

Power spectra of piston for a single delta-function layer, normalized in ordinate and abscissa as described in Eq. (71), to give Hφ(x). Overall tilt has been left in, and the segment location ranges from center to edge of telescope. Curves for both a finite segment (d = D/10) and an infinitesimally small segment (d = 0) are shown.

FIG. 5
FIG. 5

Normalized power spectra Hϕ(x) of piston for overall tilt removed. Otherwise conditions same as Fig. 4.

FIG. 6
FIG. 6

Power spectra of the azimuthal component of tilt for a single delta-function layer, normalized in ordinate and abscissa as described in Eq. (73), to give Hα(x). Both the case of overall tilt left in (n = 0) and removed (n = 1) are considered. Segment location ranges from telescope center to edge, and the segment sizes are d = D/10 and d = 0. Note that segment location and overall tilt have an effect in low values of x, whereas the size of the segment determines the high-x roll-off.

FIG. 7
FIG. 7

Normalized power spectra Hα(x) of the radial component of tilt. Otherwise conditions same as Fig. 6.

FIG. 8
FIG. 8

Power-law models for piston power spectra Hα(x). The particular case shown here is for r1/D = 0.15 so that the reader can compare with similar results in Figs. 4 and 5.

FIG. 9
FIG. 9

Power-law models for both the azimuthal and radial tilt power spectra Hα(x). The particular case shown is radial tilt for r1/D = 0.15 so that the reader can compare with similar results in Fig. 7.

FIG. 10
FIG. 10

Functions Gφ(y) Gα(y) which filter the modeled power spectra Hϕ(x) and Hα(x) and depend upon the segment size according to the new normalized frequency y = dx/D. The models for the filters are the straight lines shown and are given in Eqs. (84) and (85).

FIG. 11
FIG. 11

Integral limits for D D { · } depicted by cross-hatched area.

FIG. 12
FIG. 12

Integral limits for d d { · } depicted by cross-hatched area.

FIG. 13
FIG. 13

Integral limits for d D { · } depicted by cross–hatched area.

Equations (108)

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ϕ ( x ˜ , t ) = φ ( x ˜ , t ) - φ ( x ˜ , t ) ¯ ¯ - n β ( x ˜ , t ) ,
G ( x ˜ , t ) ¯ ¯ = A D - 1 d x ˜ W D ( x ˜ ) G ( x ˜ , t ) ,
W D ( x ˜ ) = { 1 , x ˜ D / 2 0 , x ˜ > D / 2
φ ( t ) = ϕ ( x ˜ , t ) ¯ = φ ( x ˜ , t ) ¯ - φ ( x ˜ , t ) ¯ ¯ - n β ( x ˜ , t ) ¯ .
G ( x ˜ , t ) ¯ = A d - 1 d x ˜ W d ( x ˜ ) G ( x ˜ , t ) ,
W d ( x ˜ ) = { 1 , x ˜ - r ˜ 1 d / 2 0 , x ˜ - r ˜ 1 > d / 2.
α ( t ) = 2 λ A d - 1 ( x - r 1 x ) [ ϕ ( x ˜ , t ) - ϕ ( x ˜ , t ) ¯ ] ¯ ,
( x - r 1 x ) ϕ ( x ˜ , t ) ¯ ¯ = ( x - r 1 x ) ¯ ϕ ( x ˜ , t ) ¯ ,
α ( t ) = 2 λ A d - 1 ( x - r 1 x ) ϕ ( x ˜ , t ) ¯ ,
α ( t ) = 2 λ A d - 1 [ ( x - r 1 x ) φ ( x ˜ , t ) ¯ - n ( x - r 1 x ) β ( x ˜ , t ) ¯ ] .
β ( x ˜ , t ) = ( 4 / D ) 2 [ x x φ ( x ˜ , t ) ¯ ¯ + y y φ ( x ˜ , t ) ¯ ¯ ] .
x ¯ = r 1 x ,
y ¯ = r 1 y ,
x 2 ¯ = ( d / 4 ) 2 + r 1 x 2 ,
x y ¯ = r 1 x r 1 y .
φ ( t ) = φ ( x ˜ , t ) ¯ - φ ( x ˜ , t ) ¯ ¯ - 16 n D - 2 r ˜ 1 · x ˜ φ ( x ˜ , t ) ¯ ¯
α ( t ) = 2 λ A d - 1 [ ( x - r 1 x ) φ ( x ˜ , t ) ¯ - ( d / D ) 2 n x φ ( x ˜ , t ) ¯ ¯ ]
C φ ( τ ) = φ ( t ) φ ( t + τ )
C α ( τ ) = α ( t ) α ( t + τ ) .
D D { G ( x ˜ , x ˜ ) } = A D - 2 d x ˜ d x ˜ W D ( x ˜ ) W D ( x ˜ ) G ( x ˜ , x ˜ ) , d d { G ( x ˜ , x ˜ ) } = A d - 2 d x ˜ d x ˜ W d ( x ˜ ) W d ( x ˜ ) G ( x ˜ , x ˜ ) ,
d D { G ( x ˜ , x ˜ ) } = A d - 1 A D - 1 d x ˜ d x ˜ W d ( x ˜ ) × W D ( x ˜ ) G ( x ˜ , x ˜ ) .
C φ ( τ ) = d d { C δ φ } + D D { C δ φ } - 2 d D { C δ φ } + n ( 4 / D ) 4 D D { ( r 1 x x + r 1 y y ) ( r 1 x x + r 1 y y ) C δ φ } - 2 n ( 4 / D ) 2 [ d D { ( r 1 x x + r 1 y y ) C δ φ } - D D { ( r 1 x x + r 1 y y ) C δ φ } ] .
C δ φ ( r ˜ , τ ) = φ ( x ˜ , t ) φ ( x ˜ , t + τ ) ,
r ˜ = x ˜ - x ˜
r ˜ = 1 2 ( x ˜ + x ˜ ) .
i j { · } = A i - 1 A j - 1 d r ˜ d r ˜ W i ( r ˜ - 1 2 r ˜ ) W j ( r ˜ + 1 2 r ˜ ) { · } ,
i j { · } = A i - 1 A j - 1 d r ˜ W i ( r ˜ - 1 2 r ˜ ) W j ( r ˜ + 1 2 r ˜ ) { · } .
C φ ( τ ) = d r ˜ C δ φ ( r ˜ , τ ) T φ ( r ˜ ) ,
T φ ( r ˜ ) = D D { 1 } + n ( 4 / D ) 4 D D { ( r 1 x x + r 1 y y ) ( r 1 x x + r 1 y y ) } + n 2 ( 4 / D ) 2 D D { r 1 x x + r 1 y y } + d d { 1 } - 2 [ d D { 1 } + n ( 4 / D ) 2 d D { r 1 x x + r 1 y y } ] .
C α ( τ ) = d r ˜ C δ φ ( r ˜ , τ ) T α ( r ˜ ) ,
T α ( r ˜ ) = ( 2 λ / A d ) 2 [ n ( d / D ) 4 D D { x x } + d d { ( x - r 1 x ) ( x - r 1 x ) } - 2 n ( d / D ) 2 d D { x ( x - r 1 x ) } ] .
S 1 ( r ˜ ) = D D { 1 } ,
S 2 ( r ˜ ) = ( 4 / D ) 4 D D { ( r 1 x x + r 1 y y ) ( r 1 x x + r 1 y y ) } + 2 ( 4 / D ) 2 D D { r 1 x x + r 1 y y } ,
S 3 ( r ˜ ) = d d { 1 } ,
S 4 ( r ˜ ) = d D { 1 } ,
S 5 ( r ˜ ) = ( 4 / D ) 2 d D { r 1 x x + r 1 y y } ,
S 6 ( r ˜ ) = ( d / D ) 4 D D { x x } ,
S 7 ( r ˜ ) = d d { ( x - r 1 x ) ( x - r 1 x ) } ,
S 8 ( r ˜ ) = ( d / D ) 2 d D { x ( x - r 1 x ) } .
T φ ( r ˜ ) = S 1 ( r ˜ ) + n S 2 ( r ˜ ) + S 3 ( r ˜ ) - 2 S 4 ( r ˜ ) - 2 n S 5 ( r ˜ ) ,
T α ( r ˜ ) = ( 2 λ / A d ) 2 [ n S 6 ( r ˜ ) + S 7 ( r ˜ ) - 2 n S 8 ( r ˜ ) ] .
S 1 ( r ) = 2 π A D { cos - 1 ( r / D ) - ( r / D ) [ 1 - ( r / D ) 2 ] 1 / 2 ,             0 r D 0 ,             D r .
S 2 ( r , ϑ ) = A D - 2 { 8 r 1 [ r 1 + r cos ( δ - ϑ ) ] { cos - 1 ( r D ) - ( r D ) [ 1 - ( r D ) 2 ] 1 / 2 } - 16 r r 1 2 3 D [ 1 - ( r D ) 2 ] 3 / 2 [ 1 + 4 cos 2 ( δ - ϑ ) ] ,             0 r D 0 ,             D r
S 3 ( r ) = 2 π A d { cos - 1 ( r d ) - ( r d ) [ 1 - ( r d ) 2 ] 1 / 2 ,             0 r d 0 ,             d r .
γ 1 = [ r 2 2 + ( D / 2 ) 2 - ( d / 2 ) 2 ] / ( r 2 D )
γ 2 = [ r 2 2 + ( d / 2 ) 2 - ( D / 2 ) 2 ] / ( r 2 d ) ,
r 2 = [ r 2 + r 1 2 + 2 r r 1 cos ( δ - ϑ ) ] 1 / 2 .
S 4 ( r , ϑ ) = A D - 1 A d - 1 { A d ,             0 r 2 D - d 2 A D π cos - 1 γ 1 + A d π cos - 1 γ 2 - r 2 d 2 ( 1 - γ 2 2 ) 1 / 2 ,             D - d 2 r 2 D + d 2 0 ,             D + d 2 r 2 .
S 5 ( r , ϑ ) = 4 r 1 A D - 2 [ r 1 + r cos ( δ - ϑ ) ] { π ,             0 r 2 D - d 2 cos - 1 γ 2 - A d γ 2 ( 1 - γ 2 2 ) 1 / 2 ,             D - d 2 r 2 D + d 2 0 ,             D + d 2 r 2 .
S 6 ( r , ϑ ) = { ( d / D ) 4 2 π 2 { cos - 1 ( r D ) - ( r D ) [ 1 - ( r D ) 2 ] 1 / 2 - 2 r 3 D [ 1 - ( r D ) 2 ] 3 / 2 ( 1 + 4 cos 2 ϑ ) } ,             0 r D 0 ,             D r    
S 7 ( r , ϑ ) = { 1 2 π 2 { cos - 1 ( r d ) - ( r d ) [ 1 - ( r d ) 2 ] 1 / 2 - 2 r 3 d [ 1 - ( r d ) 2 ] 3 / 2 ( 1 + 4 cos 2 ϑ ) } ,             0 r d 0 ,             d r
S 8 ( r , ϑ ) = 1 4 π 2 ( d D ) 4 { π ,             0 r 2 D - d 2 ( D d ) 4 cos - 1 γ 1 + cos - 1 γ 2 - [ γ 2 + γ 1 ( D d ) 3 ] ( 1 - γ 2 2 ) 1 / 2 - 4 r 2 3 d ( 1 - γ 2 2 ) 3 / 2 ( 1 + 4 r 2 2 ( r cos ϑ + r 1 cos δ ) 2 ) ,             D - d 2 r 2 D + d 2 0 ,             D + d 2 r 2 .
lim d 0 S 3 = δ ( r ) ,
lim d 0 S 4 = A D - 1 { 1 , 0 r 2 1 2 D 0 , 1 2 D < r 2
lim d 0 S 5 = A D - 1 ( 4 / D ) 2 r 1 [ r 1 + r cos ( δ - ϑ ) ] { 1 , 0 r 2 1 2 D 0 , 1 2 D < r 2 .
lim d 0 C α ( τ ) = lim d 0 ( 2 λ A d ) 2 d r ˜ C δ φ ( r ˜ , τ ) S 7 ( r ˜ ) = 1 2 π k 2 lim r 0 0 2 π d ϑ C δ φ ( r , ϑ , τ ) ( 1 - 4 cos 2 ϑ ) ,             ( n = 0 ) .
C δ φ ( 0 , ϑ , τ ) = lim r 0 1 r C δ φ ( r , ϑ , τ ) .
D δ φ ( r ˜ , τ ) = [ φ ( x ˜ , t ) - φ ( x ˜ + r ˜ , t ) ] [ φ ( x ˜ , t + τ ) - φ ( x ˜ + r ˜ , t + τ ) ] .
C δ φ ( r ˜ , τ ) = C δ φ ( 0 , τ ) - 1 2 D δ φ ( r ˜ , τ ) .
C φ ( τ ) = C δ φ ( 0 , τ ) d r ˜ T φ ( r ˜ ) - 1 2 d r ˜ D δ φ ( r ˜ , τ ) T φ ( r ˜ ) .
d r ˜ T φ ( r ˜ ) = 0
d r ˜ T α ( r ˜ ) = 0.
C ( φ α ) ( τ ) = - 1 2 d r ˜ D δ φ ( r ˜ , τ ) T ( φ α ) ( r ˜ ) .
W δ φ ( r ˜ , f ) = 4 0 d τ cos ( 2 π f τ ) D δ φ ( r ˜ , τ ) .
F ( φ α ) ( f ) = 4 0 d τ cos ( 2 π f τ ) C ( φ α ) ( τ ) .
F ( φ α ) ( f ) = - 1 2 d r ˜ W δ φ ( r ˜ , f ) T ( φ α ) ( r ˜ ) .
lim d 0 F α ( f ) = 1 4 π k 2 lim r 0 0 2 π d ϑ W δ φ ( r , ϑ , f ) ( 4 cos 2 ϑ - 1 ) ,             ( n = 0 )
D δ φ ( r ˜ , 0 ) = D φ ( r ˜ ) = [ φ ( x ˜ ) - φ ( x ˜ + r ˜ ) ] 2 ,
σ ( φ α ) 2 = C ( φ α ) ( 0 ) = - 1 2 d r ˜ D φ ( r ˜ ) T ( φ α ) ( r ˜ ) .
lim d 0 σ α 2 = 1 4 π k 2 lim r 0 0 2 π d ϑ D φ ( r , ϑ ) ( 4 cos 2 ϑ - 1 ) .
W δ φ ( r ˜ , f ) = 0 L d z C n 2 ( z ) M ( r ˜ , f , z ; v ˜ , λ , L ) ,
F ( φ α ) ( f ) = 0 L d z C n 2 ( z ) × ( - 1 2 d r ˜ T ( φ α ) ( r ˜ ) M ( r ˜ , f , z ; v ˜ , λ , L ) ) .
C n 2 ( z ) = i = 1 m C n i 2 δ ( s - s o i ) .
s o i - ( 1 / 2 ) Δ s i s o i + ( 1 / 2 ) Δ s i C n 2 ( s ) d s = C n i 2 Δ s i .
W δ φ ( r , f ) = 2.079 ( f / f 0 ) - 8 / 3 f 0 - 1 × ( D / r 0 ) 5 / 3 sin 2 ( f u / f 0 ) ,
f 0 = v / ( π D s ) , u = r / D ,
s = { 1 for plane waves s for spherical waves .
D φ ( r ) = 6.88 ( r / r 0 ) 5 / 3
r 0 - 5 / 3 = 0.423 k 2 L C n 2 s 5 / 3 .
r 0 ( m ) - 5 / 3 = i = 1 m r 0 i - 5 / 3 ,
lim r 0 W δ φ ( r , f ) = 4.156 ( f / f 0 ) - 2 / 3 × f 0 - 1 ( D / r 0 ) 5 / 3 D - 2 ,
lim r 0 D φ ( r ) = 9.239 r 0 - 5 / 3 ( s κ m ) 1 / 3 .
σ φ av 2 = 1.075 ( D / r 0 ) 5 / 3 ( piston only ) σ φ av 2 = 0.141 ( D / r 0 ) 5 / 3 ( piston with tilt ) .
lim d 0 σ α 2 ( d ) = 4.620 k - 2 r 0 - 5 / 3 ( s κ m ) 1 / 3 .
σ α 2 ( d ) = 0.184 ( d / r 0 ) 5 / 3 ( λ / d ) 2 ,             ( d l 0 ,             n = 0 ) .
H φ ( x ) = F φ ( f ) 0.481 f 0 ( r 0 / D ) 5 / 3
lim d 0 H φ ( x ) = 1 4 x - 8 / 3 ,             x 1.
H α ( x ) = F α ( f ) 0.481 f 0 ( r 0 / D ) 5 / 3 ( D / λ ) 2 ,
H α ( x ) = 0.02533 x - 2 / 3 .
H φ ( x ) = c 1 x - 2 / 3 , x x 1 H φ ( x ) = c 2 x 4 / 3 , x 1 x x 2 H φ ( x ) = c 3 x - 8 / 3 , x 2 x .
H φ ( x ) = c 1 x - 2 / 3 , x x 3 H φ ( x ) = c 3 x - 8 / 3 , x 3 x
c 1 = ( 1 - n ) ( r 1 / D ) 2 , c 2 { [ ( r 1 / D ) 2 - 1 12 ] 2 + ( 3 π 4 ) - 1 } , c 3 = 1 4 ,
x 1 = ( c 1 / c 2 ) 1 / 2 ,             x 2 = ( c 3 / c 2 ) 1 / 4 ,             and x 3 = ( c 3 / c 1 ) 1 / 2 .
H α ( x ) = c 4 x - 2 / 3 ;
H α ( x ) = c 5 x 4 / 3 , x x 4 H α ( x ) = c 6 x 10 / 3 , x 4 x x 5 H α ( x ) = c 4 x - 2 / 3 , x 5 x ;
H α ( x ) = c 5 x 4 / 3 , x x 6 H α ( x ) = c 4 x - 2 / 3 , x 6 x
c 4 = ( 2 π ) - 2 , c 5 = ( r 1 / D ) 2 π 2 { 1 azimuthal tilt 1 3 radial tilt , c 6 = 2.758 × 10 - 4 ,
x 4 = ( c 5 / c 6 ) 1 / 2 ,             x 5 = ( c 4 / c 6 ) 1 / 4 ,             and             x 6 = ( c 4 / c 5 ) 1 / 2 .
G φ ( y ) = { 1 ,             0 y 0.366 1.26 - 0.712 y ,             0.366 y 1.77 0 ,             1.77 y .
G α ( y ) = { 1 ,             0 y 0.332 1.12 - 0.361 y ,             0.332 y 3.10 0 ,             3.10 y .
( x x ) = p cos ϑ - q sin ϑ ± 1 2 r cos ϑ , ( y y ) = p sin ϑ + q cos ϑ ± 1 2 r sin ϑ ,
D D { · } = A D - 2 { 0 ( 1 / 2 ) ( D - r ) d p - [ ( D / 2 ) 2 - ( p + r / 2 ) 2 ] 1 / 2 [ ( D / 2 ) 2 - ( p + r / 2 ) 2 ] 1 / 2 d q + ( 1 / 2 ) ( r - D ) 0 d p - [ ( D / 2 ) 2 - ( p - r / 2 ) 2 ] 1 / 2 [ ( D / 2 ) 2 - ( p - r / 2 ) 2 ] 1 / 2 d q ,             0 r D 0 ,             D r .
ξ = p - r 1 cos μ , η = q - r 1 sin μ , ( x x ) = ξ cos ϑ - η sin ϑ + r 1 cos δ ± 1 2 r cos ϑ , ( y y ) = ξ sin ϑ + η cos ϑ + r 1 sin δ ± 1 2 r sin ϑ .
( x x ) = r 2 - 1 [ p ( r cos ϑ + r 1 cos δ ) - q ( r sin ϑ + r 1 sin δ ) ] - ( 0 r cos ϑ ) , ( y y ) = r 2 - 1 [ p ( r sin ϑ + r 1 sin δ ) + q ( r cos ϑ + r 1 cos δ ) ] - ( 0 r sin ϑ ) ,
r 2 = ( r 2 + r 1 2 + 2 r r 1 cos μ ) 1 / 2
d D { · } = A D - 1 A d - 1 × { r 2 - d / 2 r 2 + d / 2 d p - [ ( d / 2 ) 2 - ( p - r 2 ) 2 ] 1 / 2 [ ( d / 2 ) 2 - ( p - r 2 ) 2 ] 1 / 2 d q ,             0 r 2 D - d 2 [ ( D / 2 ) 2 - ( d / 2 ) 2 + r 2 2 ] / ( 2 r 2 ) D / 2 d p - [ ( D / 2 ) 2 - p 2 ] 1 / 2 [ ( D / 2 ) 2 - p 2 ] 1 / 2 d q + r 2 - d / 2 [ ( D / 2 ) 2 - ( d / 2 ) 2 + r 2 2 ] / ( 2 r 2 ) d p - [ ( d / 2 ) 2 - ( r 2 - p ) 2 ] 1 / 2 [ ( d / 2 ) 2 - ( r 2 - p ) 2 ] 1 / 2 d q , D - d 2 r 2 D + d 2 0 ,             D + d 2 r 2 .
D D { 1 } = d d { 1 } = d D { 1 } = 1 , D D { x } = D D { y } = D D { x x } = D D { y y } = D D { x y + x y } = d D { x } = d D { y } = d D { x x } = 0 , d d { x x } = r 1 x 2 , d d { x } = d d { x } = r 1 x .
d r ˜ T φ ( r ˜ ) = 0 = d r ˜ T α ( r ˜ ) .