Abstract

This work develops the concept of “diffractive anisoplanatism,” a phenomenon that limits tracker performance for directed-energy applications by introducing differences between point-source-beacon tilt measurements and scoring-beam centroid motion. Our theoretical analysis of this phenomenon, checked against wave-optics simulations, highlights two relevant effects: diffractive conversion of phase to amplitude in the beacon light, and diffractive spreading of the scoring beam into regions outside of the geometric cone sampled by the beacon. In this work, we derive expressions for the variance and power spectral density of the differential jitter between beacon tilt and scoring-beam centroid motion. Additionally, we find a scenario-dependent frequency fS above which corrections of atmospheric tilt will increase, rather than decrease, scoring-beam centroid jitter on target. Diffractive anisoplanatism provides a fundamental-physics limit on tracker performance that should be considered alongside other practical limitations.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. D. L. Fried and H. T. Yura, “Telescope-performance reciprocity for propagation in a turbulent medium,” J. Opt. Soc. Am. 62, 600–602 (1972).
    [Crossref]
  2. V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982).
    [Crossref]
  3. R. R. Parenti, J. M. Roth, J. H. Shapiro, F. G. Walther, and J. A. Greco, “Experimental observations of channel reciprocity in single-mode free-space optical links,” Opt. Express 20, 21635–21644 (2012).
    [Crossref]
  4. J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15, 022401 (2013).
    [Crossref]
  5. P. H. Merritt, S. Peterson, R. Telgarsky, S. D. O’Keefe, R. Pringle, and R. L. Brunson, “Performance of tracking algorithms under airborne turbulence,” Proc. SPIE 4376, 99–106 (2001).
    [Crossref]
  6. P. H. Merritt, S. Peterson, R. Telgarsky, R. Pringle, R. L. Brunson, and S. D. O’Keefe, “Limitation on the bandwidth of tracking through the atmosphere,” Proc. SPIE 4724, 37–44 (2002).
    [Crossref]
  7. P. Merritt and M. Spencer, Beam Control for Laser Systems, 2nd ed. (DEPS, 2018).
  8. J. P. Siegenthaler, “Guidelines for adaptive-optic correction based on aperture filtration,” Ph.D. thesis (University of Notre Dame, 2008).
  9. S. Enguehard and B. Hatfield, “Introduction to laser guide star theory versus experiment,” Proc. SPIE 5895, 34–38 (2005).
    [Crossref]
  10. S. E. J. Shaw and E. M. Tomlinson, “Analytic propagation variances and power spectral densities from a wave-optics perspective,” J. Opt. Soc. Am. A 36, 1267–1278 (2019).
    [Crossref]
  11. M. C. Roggemann and B. Welch, Imaging through Turbulence (CRC Press, 1996).
  12. G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A 11, 358–367 (1994).
    [Crossref]
  13. D. P. Greenwood, “Bandwidth specification for adaptive optics systems,” J. Opt. Soc. Am. 67, 390–393 (1977).
    [Crossref]
  14. H. T. Yura and M. T. Tavis, “Centroid anisoplanatism,” J. Opt. Soc. Am. A 2, 765–773 (1985).
    [Crossref]
  15. C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24, 144–154 (1976).
    [Crossref]
  16. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, 2nd ed. (SPIE, 2007).

2019 (1)

2013 (1)

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15, 022401 (2013).
[Crossref]

2012 (1)

2005 (1)

S. Enguehard and B. Hatfield, “Introduction to laser guide star theory versus experiment,” Proc. SPIE 5895, 34–38 (2005).
[Crossref]

2002 (1)

P. H. Merritt, S. Peterson, R. Telgarsky, R. Pringle, R. L. Brunson, and S. D. O’Keefe, “Limitation on the bandwidth of tracking through the atmosphere,” Proc. SPIE 4724, 37–44 (2002).
[Crossref]

2001 (1)

P. H. Merritt, S. Peterson, R. Telgarsky, S. D. O’Keefe, R. Pringle, and R. L. Brunson, “Performance of tracking algorithms under airborne turbulence,” Proc. SPIE 4376, 99–106 (2001).
[Crossref]

1994 (1)

1985 (1)

1982 (1)

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982).
[Crossref]

1977 (1)

1976 (1)

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24, 144–154 (1976).
[Crossref]

1972 (1)

Brunson, R. L.

P. H. Merritt, S. Peterson, R. Telgarsky, R. Pringle, R. L. Brunson, and S. D. O’Keefe, “Limitation on the bandwidth of tracking through the atmosphere,” Proc. SPIE 4724, 37–44 (2002).
[Crossref]

P. H. Merritt, S. Peterson, R. Telgarsky, S. D. O’Keefe, R. Pringle, and R. L. Brunson, “Performance of tracking algorithms under airborne turbulence,” Proc. SPIE 4376, 99–106 (2001).
[Crossref]

Butts, R.

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24, 144–154 (1976).
[Crossref]

Charnotskii, M. I.

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982).
[Crossref]

Dolfi, D.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15, 022401 (2013).
[Crossref]

Enguehard, S.

S. Enguehard and B. Hatfield, “Introduction to laser guide star theory versus experiment,” Proc. SPIE 5895, 34–38 (2005).
[Crossref]

Fried, D. L.

Greco, J. A.

Greenwood, D. P.

Hatfield, B.

S. Enguehard and B. Hatfield, “Introduction to laser guide star theory versus experiment,” Proc. SPIE 5895, 34–38 (2005).
[Crossref]

Hogge, C.

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24, 144–154 (1976).
[Crossref]

Lukin, V. P.

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982).
[Crossref]

Merritt, P.

P. Merritt and M. Spencer, Beam Control for Laser Systems, 2nd ed. (DEPS, 2018).

Merritt, P. H.

P. H. Merritt, S. Peterson, R. Telgarsky, R. Pringle, R. L. Brunson, and S. D. O’Keefe, “Limitation on the bandwidth of tracking through the atmosphere,” Proc. SPIE 4724, 37–44 (2002).
[Crossref]

P. H. Merritt, S. Peterson, R. Telgarsky, S. D. O’Keefe, R. Pringle, and R. L. Brunson, “Performance of tracking algorithms under airborne turbulence,” Proc. SPIE 4376, 99–106 (2001).
[Crossref]

Minet, J.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15, 022401 (2013).
[Crossref]

O’Keefe, S. D.

P. H. Merritt, S. Peterson, R. Telgarsky, R. Pringle, R. L. Brunson, and S. D. O’Keefe, “Limitation on the bandwidth of tracking through the atmosphere,” Proc. SPIE 4724, 37–44 (2002).
[Crossref]

P. H. Merritt, S. Peterson, R. Telgarsky, S. D. O’Keefe, R. Pringle, and R. L. Brunson, “Performance of tracking algorithms under airborne turbulence,” Proc. SPIE 4376, 99–106 (2001).
[Crossref]

Parenti, R. R.

Peterson, S.

P. H. Merritt, S. Peterson, R. Telgarsky, R. Pringle, R. L. Brunson, and S. D. O’Keefe, “Limitation on the bandwidth of tracking through the atmosphere,” Proc. SPIE 4724, 37–44 (2002).
[Crossref]

P. H. Merritt, S. Peterson, R. Telgarsky, S. D. O’Keefe, R. Pringle, and R. L. Brunson, “Performance of tracking algorithms under airborne turbulence,” Proc. SPIE 4376, 99–106 (2001).
[Crossref]

Polnau, E.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15, 022401 (2013).
[Crossref]

Pringle, R.

P. H. Merritt, S. Peterson, R. Telgarsky, R. Pringle, R. L. Brunson, and S. D. O’Keefe, “Limitation on the bandwidth of tracking through the atmosphere,” Proc. SPIE 4724, 37–44 (2002).
[Crossref]

P. H. Merritt, S. Peterson, R. Telgarsky, S. D. O’Keefe, R. Pringle, and R. L. Brunson, “Performance of tracking algorithms under airborne turbulence,” Proc. SPIE 4376, 99–106 (2001).
[Crossref]

Roggemann, M. C.

M. C. Roggemann and B. Welch, Imaging through Turbulence (CRC Press, 1996).

Roth, J. M.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, 2nd ed. (SPIE, 2007).

Shapiro, J. H.

Shaw, S. E. J.

Siegenthaler, J. P.

J. P. Siegenthaler, “Guidelines for adaptive-optic correction based on aperture filtration,” Ph.D. thesis (University of Notre Dame, 2008).

Spencer, M.

P. Merritt and M. Spencer, Beam Control for Laser Systems, 2nd ed. (DEPS, 2018).

Tavis, M. T.

Telgarsky, R.

P. H. Merritt, S. Peterson, R. Telgarsky, R. Pringle, R. L. Brunson, and S. D. O’Keefe, “Limitation on the bandwidth of tracking through the atmosphere,” Proc. SPIE 4724, 37–44 (2002).
[Crossref]

P. H. Merritt, S. Peterson, R. Telgarsky, S. D. O’Keefe, R. Pringle, and R. L. Brunson, “Performance of tracking algorithms under airborne turbulence,” Proc. SPIE 4376, 99–106 (2001).
[Crossref]

Tomlinson, E. M.

Tyler, G. A.

Vorontsov, M. A.

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15, 022401 (2013).
[Crossref]

Walther, F. G.

Welch, B.

M. C. Roggemann and B. Welch, Imaging through Turbulence (CRC Press, 1996).

Yura, H. T.

IEEE Trans. Antennas Propag. (1)

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24, 144–154 (1976).
[Crossref]

J. Opt. (1)

J. Minet, M. A. Vorontsov, E. Polnau, and D. Dolfi, “Enhanced correlation of received power-signal fluctuations in bidirectional optical links,” J. Opt. 15, 022401 (2013).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Express (1)

Proc. SPIE (3)

S. Enguehard and B. Hatfield, “Introduction to laser guide star theory versus experiment,” Proc. SPIE 5895, 34–38 (2005).
[Crossref]

P. H. Merritt, S. Peterson, R. Telgarsky, S. D. O’Keefe, R. Pringle, and R. L. Brunson, “Performance of tracking algorithms under airborne turbulence,” Proc. SPIE 4376, 99–106 (2001).
[Crossref]

P. H. Merritt, S. Peterson, R. Telgarsky, R. Pringle, R. L. Brunson, and S. D. O’Keefe, “Limitation on the bandwidth of tracking through the atmosphere,” Proc. SPIE 4724, 37–44 (2002).
[Crossref]

Sov. J. Quantum Electron. (1)

V. P. Lukin and M. I. Charnotskii, “Reciprocity principle and adaptive control of optical radiation parameters,” Sov. J. Quantum Electron. 12, 602–605 (1982).
[Crossref]

Other (4)

P. Merritt and M. Spencer, Beam Control for Laser Systems, 2nd ed. (DEPS, 2018).

J. P. Siegenthaler, “Guidelines for adaptive-optic correction based on aperture filtration,” Ph.D. thesis (University of Notre Dame, 2008).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, 2nd ed. (SPIE, 2007).

M. C. Roggemann and B. Welch, Imaging through Turbulence (CRC Press, 1996).

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Figures (6)

Fig. 1.
Fig. 1. Illustration of the propagation of point-source beacon light to an aperture (yellow cone) and a scoring beam to the target plane (red region), showing the two diffractive effects that lead to anisoplanatism. First is the conversion of beacon phase into amplitude, indicated by a change in color in the yellow cone, which results in a loss of phase information on the beacon light. Second is the spreading of the focused scoring beam outside of the cone sampled by the beacon, spoiling the geometric overlap of the two beams.
Fig. 2.
Fig. 2. These plots illustrate the breakdown of tilt reciprocity between beacon G tilt and Z tilt and scoring-beam centroid motion, all due to single sinusoidal phase profiles of varying κ . The plots are of Eqs. (3)–(5), with the shared cosine terms removed because all three signals have identical time dependence. Our sinusoidal phase profiles have unit amplitude and are placed at the midpoint of a 3-km path, the fields have 1-μm wavelengths, and we have a 20-cm aperture. In panel (a), we show a comparison with D S ( z ) = D B ( z ) = ( 1 z / L ) · 20 cm , so the loss of tilt reciprocity is due entirely to diffractive conversion of beacon phase to amplitude. In panel (b), we take D S ( z ) = D B ( z ) + ( z / L ) · 1 cm to simulate the effect of diffractive spreading of the scoring beam. In panel (c), we take the differential tilts to show how the measurements increasingly differ in magnitude, sometimes in sign, as κ increases.
Fig. 3.
Fig. 3. These plots shows a comparison between the PSDs calculated with our analytic expressions for scoring-beam centroid jitter in Eq. (11) and beacon G tilt in Eq. (12), as well as results of wave-optics modeling, shown for m ^ v in panel (a) and for m ^ v in panel (b). The theoretical curves agree well with wave optics, and both show the predicted disagreement between beacon and scoring beam at high frequencies. The scenario chosen was propagation over a 3-km path with a constant 0.2 m/s cross wind, Rytov variance of 0.01, constant C n 2 4 × 10 16 m 2 / 3 , and Kolmogorov turbulence. The scoring beam is a 20-cm-diameter top-hat beam at 1-μm wavelength, and the beacon is a point source at the same wavelength and measured in the same aperture. With these parameters, the characteristic frequency v / D is 1 Hz. We have modeled weak turbulence here to show that diffractive anisoplanatism is not a strong-turbulence phenomenon; in the weak-turbulence regime, each PSD is linear in C n 2 .
Fig. 4.
Fig. 4. In this plot, we compare the variance of differential jitter in Eq. (15) to the scoring-beam centroid jitter and beacon G tilt in Eqs. (16) and (17). The calculations were performed for the same scenario used in Fig. 3, except that the aperture diameter was varied to change the Fresnel number of the propagation. The residual jitter variance is linear in the turbulence strength, and thus easily scaled from these values.
Fig. 5.
Fig. 5. This plot shows the comparison between the PSDs of scoring-beam centroid jitter, beacon G tilt, and the differential jitter between them, calculated for the same scenario used in Fig. 3. We show the two-axis jitter, found by summing the PSDs parallel and perpendicular to the wind. We see that there is a frequency f S 12 Hz above which the differential jitter [yellow curve, from Eq. (18)] has more power than the scoring-beam motion [blue curve, from Eq. (11)], meaning corrections above that frequency will increase jitter on target. At frequencies above f S , the differential-jitter curve lies on top of the beacon G tilt [red curve, from Eq. (12)]. The characteristic frequency v / D is again 1 Hz.
Fig. 6.
Fig. 6. The plot in panel (a) shows the calculated f S values for a wide range of scenarios with varying aperture size, wind speed, and Fresnel number, with v / D ranging from 0.01 to 100 Hz and Fresnel number ranging from 0.3 to 100, but all with constant C n 2 and wind speed along the path. We see good agreement with our scaling law over more than 6 orders of magnitude of f S , with an RMS error of less than 7%. When we consider cases with variable wind speed (up to 10 × along the path) and C n 2 (up to 100 × along the path), or Fresnel numbers less than 0.3, we see deviation from the scaling law; 95% of the cases run lie within the bounds indicated in panel (b) and still clearly follow the trend of the scaling law when we use the maximum wind speed along the path.

Equations (19)

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ϕ ( κ , r , z , t ) = sin [ κ · r + α ( κ , z ) κ · v ( z ) t ] .
1 L X m ( κ , z , t ; L ) = ( L z ) κ L k cos [ α ( κ , z ) κ · v ( z ) t ] × cos ( ϕ m ϕ κ ) F [ a 2 ( r , z ) P ] ,
θ G , m ( κ , z , t ; 0 ) = 4 k D B ( 0 ) cos [ α ( κ , z ) κ · v ( z ) t ] × cos ( ϕ m ϕ κ ) cos ( κ 2 D B ( z ) z 2 k D B ( 0 ) ) J 1 ( D B ( z ) κ 2 ) ,
θ Z , m ( κ , z , t ; 0 ) = 32 k κ D B ( z ) D B ( 0 ) cos [ α ( κ , z ) κ · v ( z ) t ] × cos ( ϕ m ϕ κ ) cos ( κ 2 D B ( z ) z 2 k D B ( 0 ) ) J 2 ( D B ( z ) κ 2 ) .
1 L X m ( κ , z , t ; L ) = 4 ( L z ) L k D S ( z ) cos [ α ( κ , z ) κ · v ( z ) t ] cos ( ϕ m ϕ κ ) × J 1 ( D S ( z ) κ 2 ) .
σ X / L , m 2 ( κ , z ; L ) = ( L z ) 2 κ 2 2 L 2 k 2 cos 2 ( ϕ m ϕ κ ) { F [ a 2 ( r , z ) P ] } 2
σ G , m 2 ( κ , z ; 0 ) = 8 k 2 D B 2 ( 0 ) cos 2 ( ϕ m ϕ κ ) cos 2 ( κ 2 D B ( z ) z 2 k D B ( 0 ) ) × J 1 2 ( D B ( z ) κ 2 )
σ 2 = 4 π k 2 0 L d z C n 2 ( z ) 0 d κ κ ϕ n 0 ( κ ) 0 2 π d ϕ κ σ 2 ( κ , z ; L ) ,
κ = 2 π f v ( z ) cos ( ϕ f ) and ϕ κ = ϕ f + ϕ v ( z ) ,
PSD ( f ) = 32 π 3 k 2 f 0 L d z C n 2 ( z ) v 2 ( z ) π / 2 π / 2 d ϕ f sec 2 ( ϕ f ) × ϕ n 0 ( 2 π f v ( z ) cos ( ϕ f ) ) × σ 2 [ 2 π f v ( z ) cos ( ϕ f ) , ϕ f + ϕ v ( z ) , z ; L ] .
PSD X / L , m ( f ) = 10 ( 2 π ) 1 / 3 9 Γ ( 1 / 3 ) f 2 / 3 0 L d z C n 2 ( z ) v 1 / 3 ( z ) ( 1 z L ) 2 × π / 2 π / 2 d ϕ f cos 1 / 3 ( ϕ f ) cos 2 [ ϕ m ϕ f ϕ v ( z ) ] × { F [ a 2 ( r , z ) P ] κ f } 2 ,
PSD G , m ( f ) = 80 2 2 / 3 9 π 5 / 3 Γ ( 1 / 3 ) D B 2 ( 0 ) f 8 / 3 × 0 L d z C n 2 ( z ) v 5 / 3 ( z ) × π / 2 π / 2 d ϕ f cos 5 / 3 ( ϕ f ) cos 2 [ ϕ m ϕ f ϕ v ( z ) ] × cos 2 [ D B ( z ) z 2 k D B ( 0 ) ( 2 π f v ( z ) cos ( ϕ f ) ) 2 ] × J 1 2 ( π f D B ( z ) v ( z ) cos ( ϕ f ) ) .
θ X / L G , m ( κ , z ; L ) = 1 L X m ( κ , z ; L ) θ G , m ( κ , z ; 0 ) ,
σ X / L G , m 2 ( κ , z ; L ) = ( L z ) 2 2 k 2 L 2 cos 2 ( ϕ κ ϕ m ) { κ F [ a 2 ( r , z ) P ] 4 D B ( z ) cos ( κ 2 D B ( z ) z 2 k D B ( 0 ) ) J 1 ( D B ( z ) κ 2 ) } 2 ,
σ X / L G , m 2 = 5 π 9 Γ ( 1 / 3 ) 0 L d z C n 2 ( z ) ( 1 z L ) 2 0 d κ κ 8 / 3 × { κ F [ a 2 ( r , z ) P ] 4 D B ( z ) cos ( κ 2 D B ( z ) z 2 k D B ( 0 ) ) × J 1 ( D B ( z ) κ 2 ) } 2 .
σ X / L , m 2 = 5 π 9 Γ ( 1 / 3 ) 0 L d z C n 2 ( z ) ( 1 z L ) 2 0 d κ κ 2 / 3 × { F [ a 2 ( r , z ) P ] } 2 ,
σ G , m 2 = 80 π 9 Γ ( 1 / 3 ) D B 2 ( 0 ) 0 L d z C n 2 ( z ) 0 d κ κ 8 / 3 × cos 2 ( κ 2 D B ( z ) z 2 k D B ( 0 ) ) J 1 2 ( D B ( z ) κ 2 ) .
PSD X / L G , m ( f ) = 5 2 2 / 3 9 π 5 / 3 Γ ( 1 / 3 ) f 8 / 3 0 L d z C n 2 ( z ) v 5 / 3 ( z ) ( 1 z L ) 2 π / 2 π / 2 d ϕ f cos 5 / 3 ( ϕ f ) cos 2 [ ϕ m ϕ f ϕ v ( z ) ] × { 2 π f v ( z ) cos ( ϕ f ) F [ a 2 ( r , z ) P ] κ f 4 D B ( z ) cos [ D B ( z ) z 2 k D B ( 0 ) ( 2 π f v ( z ) cos ( ϕ f ) ) 2 ] J 1 ( π f D B ( z ) v ( z ) cos ( ϕ f ) ) } 2 .
f S 0.945 v D L λ = 3.78 v D N F ,