Abstract

A phase screen ribbon extrusion process is presented that allows a phase screen ribbon of any specified width to be extruded, one column at a time, producing a ribbon of any desired length, with Kolmogorov statistics (i.e., having a five-thirds power-law-dependent structure function) for all separations up to some selected upper limit—which upper limit can be as large as desired. The method is an adaptation of the method described by [ Assémat et al.Opt. Express 14, 988 (2006) ].

© 2008 Optical Society of America

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References

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  1. F. Assémat, R. W. Wilson, and E. Gendron, "Method for simulating infinitely long and non stationary phase screens with optimized memory storage," Opt. Express 14, 988-999 (2006).
    [CrossRef] [PubMed]
  2. P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, 1979), Vol. 1, p. 110, Eqs. (3-111) and (3-112). We are indebted to David Iny of Northrop-Grumman for calling our attention to this.
  3. Ref. , p. 111, Eq. (3-114).
  4. A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic, 1970), p. 33.
  5. B. Noble and J. W. Daniel, Applied Linear Algebra, 3rd ed. (Prentice-Hall, 1988), p. 418, Theorem 10.25.
  6. R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
    [CrossRef]

2006 (1)

1992 (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Opt. Express (1)

Waves Random Media (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
[CrossRef]

Other (4)

P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, 1979), Vol. 1, p. 110, Eqs. (3-111) and (3-112). We are indebted to David Iny of Northrop-Grumman for calling our attention to this.

Ref. , p. 111, Eq. (3-114).

A. H. Jazwinski, Stochastic Processes and Filtering Theory (Academic, 1970), p. 33.

B. Noble and J. W. Daniel, Applied Linear Algebra, 3rd ed. (Prentice-Hall, 1988), p. 418, Theorem 10.25.

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Figures (7)

Fig. 1
Fig. 1

Sample stencil pattern.

Fig. 2
Fig. 2

Phase screen ribbon structure function results for a 257-element wide ribbon. (a) Tenth extruded section, starting with an all-zeros initial section. (b) Fourth extruded section, starting with an initial section filled using our version of the Lane et al. algorithm. Each plot shows three curves. In each the (barely visible) dotted line shows the five-thirds, power-law, Kolmogorov theory values. The dashed line and the solid line show mean square difference values for separations along the length and for separations along the width of the phase screen ribbon, respectively. These mean values are each calculated from 1,000 independent realizations of a 257 × 1028 section of the phase screen ribbon. The stencil pattern used in developing these results had a total length equal to four times the width of the phase screen ribbon.

Fig. 3
Fig. 3

Row dependence for various separations. The same set of phase screen ribbon sections as was used to produce the results shown in Fig. 2 have been processed, treating each row separately to produce the results shown here. The ten solid curves show the mean square difference values for separations of 2, 4, 8 , , and 1,024 along the direction of the length of the ribbon. Accompanying each such curve there is a (just barely visible) dotted horizontal line showing the Kolmogorov theory value.

Fig. 4
Fig. 4

Row dependence for a separation of two units. Shown here is a limited portion of the lowest of the solid line curves shown in Fig. 3. The expanded (and linear) scale of the ordinate allows a row-dependent variation in the mean square difference to be clearly seen. The row-to-row sawtooth variation appears to have an amplitude of no more than about 2.3%. (Note: what appears as a sawtooth pattern could just as well be represented as a square-wave pattern. That it appears as a sawtooth is due to there being no data points to be plotted between the successive max and min values.)

Fig. 5
Fig. 5

Long phase screen ribbon mean square difference results. The solid curve represents the mean of the square of the difference of phase screen ribbon values, the average being formed over N RR = 2,630 independent random realizations of the phase screen ribbon section. The separations considered are aligned with the length of the ribbon, and for each separation the averaging process utilizes every pair of positions with that separation. The ribbon width is 257, and the length of the section is 100 × 257 = 25,700 . Also shown here is a (barely discernable) dotted line indicating the five-thirds power-law dependence the mean square difference should follow.

Fig. 6
Fig. 6

Normalized deviation of long phase screen ribbon sections. The solid curve shows the normalized deviation of the mean square difference results shown in Fig. 5, i.e., the difference between the mean square values and the five-thirds power-law values, divided by the five-thirds power-law values. The dashed curve shows the rms value that is to be associated with the normalized deviation. These results are for N RR = 2,630 random realizations of a phase screen ribbon section of size 257 × 25,700 . The separations are all parallel to the length of the ribbon.

Fig. 7
Fig. 7

Mean square difference results for separations much larger than the stencil length. The small filled circles show the calculated average of the squared difference of phase screen values. A 6.88 ( separation ) 5 3 dependence, i.e., a Kolmogorov dependence, is shown by the solid line. The dashed line shows a linear dependence on separation. The left of the two faint vertical dotted lines is at a separation of 1,281—which is the length of the stencil used in generating the phase screen ribbons. It is to be noted that the structure function results are very close to Kolmogorov predictions for separations up to about 4,000. The right of the two vertical dotted lines shows where the Kolmogorov and the linear dependence lines intersect, which is at a separation of 7,549.

Equations (7)

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P xx = xx T , P zz = zz T , P xz = xz T , P zx = zx T .
x = Az + Bg ,
A = P xz P zz 1 , B = P xx P xz P zz 1 P zx ,
x = [ A ( z z Ref ) + Bg ] + z Ref ,
( α γ ) ( β γ ) = 1 2 ( α β ) 2 + 1 2 ( α γ ) 2 + 1 2 ( β γ ) 2 .
σ 2 = 2 N RR N Pairs + ( 1 2 N RR 1 2 N RR N Pairs ) V ̃ ( p P , q Q ) ,
V ̃ ( μ , ν ) = [ μ 2 + ν 2 ] 5 3 ( 1 μ ) + ( 1 μ ) d x ( ζ ν ) + ( ζ ν ) d y [ 1 μ x ( 1 μ ) 2 ] [ ζ ν y ( ζ ν ) 2 ] ( [ ( x + μ ) 2 + ( y + ν ) 2 ] 5 6 + [ ( x μ ) 2 + ( y ν ) 2 ] 5 6 2 [ x 2 + y 2 ] 5 6 ) 2 .

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