Abstract

Phase compensation instability (PCI) is a runaway process associated with the operation of a closed-loop phase-only adaptive-optics system transmitting through the atmosphere a large-diameter laser beam with a high power density. Under most conditions such an operation introduces positive feedback;  if the laser power density is sufficiently high, there is the possibility of PCI-type runaway of the adaptive-optics servo. It is now known that the occurrence of PCI is inhibited by wind shear (wind-shear induced stabilization of PCI, or WISP) because at every different position along the propagation path the turbulent wind has a (slightly) different component of velocity in the plane transverse to the laser beam’s propagation direction. We develop a set of equations from which the conditions for the onset of PCI can be determined. Sample results are presented for propagation through an atmosphere with a negative exponential absorption profile—which nominally corresponds to ground-to-space propagation. The results indicate that for typical conditions (as a consequence of wind shear) one can transmit a rather substantial laser power density without encountering PCI. As an example of the application of our theory, the expected Strehl ratio for a HV5/7 turbulence model is evaluated for the parameters of a nominal ground-to-space high-energy laser beam propagation system.

© 1998 Optical Society of America

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References

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  1. T. J. Karr, “Thermal blooming compensation instabilities,” J. Opt. Soc. Am. A 6, 1038–1048 (1989).
    [CrossRef]
  2. S. Enguehard, B. Hatfield, “Perturbative approach to the small-scale physics of the interaction of thermal blooming and turbulence,” J. Opt. Soc. Am. A 8, 637–646 (1991).
    [CrossRef]
  3. J. F. Schonfeld, “Linearized theory of thermal-blooming phase compensation instability with realistic adaptive-optics geometry,” J. Opt. Soc. Am. B 9, 1803–1812 (1992).
    [CrossRef]
  4. V. Lukin, B. Fortes, “The influence of wavefront dislocations on phase conjugation instability at thermal blooming compensation,” Pure Appl. Opt. 6, 103–116 (1997).
    [CrossRef]
  5. D. G. Fouche, C. Higgs, C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” The Lincoln Laboratory J. 5, 273–292 (1992).
  6. D. L. Fried, “Blurring effects in thermal blooming,” (Optical Sciences Company, Anaheim, Calif., August1988).
  7. D. L. Fried, R. K. Szeto, “Turbulence and thermal blooming interaction (TTBI): static,” (Optical Sciences Company, Anaheim, Calif., September1988).
  8. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).
  9. R. K. Szeto, D. L. Fried, “Mini-shear effects and PCI,” (Optical Sciences Company, Anaheim, Calif., July1990).
  10. H. B. Keller, “A new difference scheme for parabolic problems,” in Numerical Solutions of Partial Differential Equations II, B. Hubbard, ed. (Academic, New York, 1971).
  11. Work is currently in preparation by R. Szeto on numerical modeling of small-scale physics in atmospheric-turbulence propagation.
  12. R. K. Szeto, D. L. Fried, “Steady state TTBI singular points and PCI thresholds,” (Optical Sciences Company, Anaheim, Calif., August1990).
  13. D. E. Novoseller, TRW, Redondo Beach, Calif. 90278 (personal communication; letter to P. Berger, copy to tOSC, dated July1, 1991).
  14. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  15. J. L. Walsh, P. B. Ulrich, “Thermal blooming in the atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978).
  16. D. L. Fried, “Analysis of turbulence velocity spread data,” (Optical Sciences Company, Anaheim, Calif., September1988).

1997 (1)

V. Lukin, B. Fortes, “The influence of wavefront dislocations on phase conjugation instability at thermal blooming compensation,” Pure Appl. Opt. 6, 103–116 (1997).
[CrossRef]

1992 (2)

D. G. Fouche, C. Higgs, C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” The Lincoln Laboratory J. 5, 273–292 (1992).

J. F. Schonfeld, “Linearized theory of thermal-blooming phase compensation instability with realistic adaptive-optics geometry,” J. Opt. Soc. Am. B 9, 1803–1812 (1992).
[CrossRef]

1991 (1)

1989 (1)

1966 (1)

Berger, P.

D. E. Novoseller, TRW, Redondo Beach, Calif. 90278 (personal communication; letter to P. Berger, copy to tOSC, dated July1, 1991).

Enguehard, S.

Fortes, B.

V. Lukin, B. Fortes, “The influence of wavefront dislocations on phase conjugation instability at thermal blooming compensation,” Pure Appl. Opt. 6, 103–116 (1997).
[CrossRef]

Fouche, D. G.

D. G. Fouche, C. Higgs, C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” The Lincoln Laboratory J. 5, 273–292 (1992).

Fried, D. L.

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
[CrossRef]

R. K. Szeto, D. L. Fried, “Steady state TTBI singular points and PCI thresholds,” (Optical Sciences Company, Anaheim, Calif., August1990).

D. L. Fried, “Analysis of turbulence velocity spread data,” (Optical Sciences Company, Anaheim, Calif., September1988).

D. L. Fried, “Blurring effects in thermal blooming,” (Optical Sciences Company, Anaheim, Calif., August1988).

D. L. Fried, R. K. Szeto, “Turbulence and thermal blooming interaction (TTBI): static,” (Optical Sciences Company, Anaheim, Calif., September1988).

R. K. Szeto, D. L. Fried, “Mini-shear effects and PCI,” (Optical Sciences Company, Anaheim, Calif., July1990).

Hatfield, B.

Higgs, C.

D. G. Fouche, C. Higgs, C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” The Lincoln Laboratory J. 5, 273–292 (1992).

Karr, T. J.

Keller, H. B.

H. B. Keller, “A new difference scheme for parabolic problems,” in Numerical Solutions of Partial Differential Equations II, B. Hubbard, ed. (Academic, New York, 1971).

Lukin, V.

V. Lukin, B. Fortes, “The influence of wavefront dislocations on phase conjugation instability at thermal blooming compensation,” Pure Appl. Opt. 6, 103–116 (1997).
[CrossRef]

Novoseller, D. E.

D. E. Novoseller, TRW, Redondo Beach, Calif. 90278 (personal communication; letter to P. Berger, copy to tOSC, dated July1, 1991).

Pearson, C. F.

D. G. Fouche, C. Higgs, C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” The Lincoln Laboratory J. 5, 273–292 (1992).

Schonfeld, J. F.

Szeto, R.

Work is currently in preparation by R. Szeto on numerical modeling of small-scale physics in atmospheric-turbulence propagation.

Szeto, R. K.

R. K. Szeto, D. L. Fried, “Steady state TTBI singular points and PCI thresholds,” (Optical Sciences Company, Anaheim, Calif., August1990).

D. L. Fried, R. K. Szeto, “Turbulence and thermal blooming interaction (TTBI): static,” (Optical Sciences Company, Anaheim, Calif., September1988).

R. K. Szeto, D. L. Fried, “Mini-shear effects and PCI,” (Optical Sciences Company, Anaheim, Calif., July1990).

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Ulrich, P. B.

J. L. Walsh, P. B. Ulrich, “Thermal blooming in the atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978).

Walsh, J. L.

J. L. Walsh, P. B. Ulrich, “Thermal blooming in the atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

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

Pure Appl. Opt. (1)

V. Lukin, B. Fortes, “The influence of wavefront dislocations on phase conjugation instability at thermal blooming compensation,” Pure Appl. Opt. 6, 103–116 (1997).
[CrossRef]

The Lincoln Laboratory J. (1)

D. G. Fouche, C. Higgs, C. F. Pearson, “Scaled atmospheric blooming experiments (SABLE),” The Lincoln Laboratory J. 5, 273–292 (1992).

Other (10)

D. L. Fried, “Blurring effects in thermal blooming,” (Optical Sciences Company, Anaheim, Calif., August1988).

D. L. Fried, R. K. Szeto, “Turbulence and thermal blooming interaction (TTBI): static,” (Optical Sciences Company, Anaheim, Calif., September1988).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

R. K. Szeto, D. L. Fried, “Mini-shear effects and PCI,” (Optical Sciences Company, Anaheim, Calif., July1990).

H. B. Keller, “A new difference scheme for parabolic problems,” in Numerical Solutions of Partial Differential Equations II, B. Hubbard, ed. (Academic, New York, 1971).

Work is currently in preparation by R. Szeto on numerical modeling of small-scale physics in atmospheric-turbulence propagation.

R. K. Szeto, D. L. Fried, “Steady state TTBI singular points and PCI thresholds,” (Optical Sciences Company, Anaheim, Calif., August1990).

D. E. Novoseller, TRW, Redondo Beach, Calif. 90278 (personal communication; letter to P. Berger, copy to tOSC, dated July1, 1991).

J. L. Walsh, P. B. Ulrich, “Thermal blooming in the atmosphere,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, New York, 1978).

D. L. Fried, “Analysis of turbulence velocity spread data,” (Optical Sciences Company, Anaheim, Calif., September1988).

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Figures (9)

Fig. 1
Fig. 1

Simple section versus complex section results: 400 sections, η0=0.5, κ˜=0.5. The rms deformable mirror command σ(τ) is plotted for both approaches. The simple section result is ragged and eventually goes exponential. The complex section result, shown by the horizontal almost perfectly straight line, attains steady state after a few t0’s and appears here as a horizontal line. (There is a small variation very near τ=0, but it is too brief to be noticeable as shown here.)

Fig. 2
Fig. 2

Effects of number of sections on results obtained with the simple section approach: η0=0.5, κ˜=0.5. The rms deformable mirror command σ(τ) is plotted for the three cases where the number of sections used is N=25, 100, and 400. Results show strong dependence on the number of sections used, indicating the spurious nature of these results.

Fig. 3
Fig. 3

Effects of number of sections on complex-section approach results: η0=0.5, κ˜=0.5. The rms deformable mirror command σ(τ) is plotted for three cases where the number of sections used is N=25, 100, and 400. There is a slight quantitative (about 3%) discrepancy between the results for the different number of sections;  this discrepancy in the steady state values can be attributed to diffraction effects not being accurately modeled for 25 sections.

Fig. 4
Fig. 4

In phase σC(τ) and quadrature phase σS(τ), components of the adaptive optics actuator commands for η0=1.5, κ˜=0.5. The quantities σC(τ) and σS(τ) correspond to GCI(κ˜, 0, τ; Z) and GSI(κ˜, 0, τ; Z), respectively. The results shown here are related to those shown in Fig. 5 by the equation σ(τ)=[σC2(τ)+σS2(τ)]1/2. The oscillatory nature of σC(τ) and of σS(τ) is not due to a physical oscillation but rather to the exponentially growing atmospheric heating pattern moving at a different velocity than the seed turbulence screen velocity V(Z) to which all results have been referenced.

Fig. 5
Fig. 5

PCI effects: complex-section approach for η0=1.5, κ˜=0.5. We see strong exponential divergence for this value of η0 and κ˜.

Fig. 6
Fig. 6

Singular points and estimates of PCI threshold. The critical WISP number for the steady-state WISP equations is shown as a function of normalized spatial frequency κ˜. The region to the left and below the curve is PCI free (i.e., the steady-state solution is bounded). The vertical bars represent the upper and lower limits within which we have been able to establish, from the study of time-dependent solutions, that the transition to exponential growth (i.e., the transition to PCI) occurs. The two circles are taken from the SABLE experimental data16: experimental PCI was observed for the circle above the curve, and experimental stable adaptive-optics operation was observed for the circle below the curve. The match of these vertical bars and the two circles from experimental data to the theoretical steady-state critical WISP number curve is obviously quite good.

Fig. 7
Fig. 7

Optical strength of turbulence CN2 for HV-5/7. The value of the refractive-index structure constant CN2 is plotted as a function of altitude for the HV5/7 turbulence model.

Fig. 8
Fig. 8

The turbulence weighted Strehl-integral kernel PAVG2(κ˜)κ˜-8/3. The CN2 weighted kernel PAVG2(κ˜)κ˜-8/3 is plotted for both the closed-loop operation (solid curves) and open-loop operation (dashed curves) as a function of κ˜. Each set has four curves corresponding to η0=0.1, 0.6, 1.0, and 1.4 with what is the bottom curve on the left corresponding to the highest values of η0 for closed-loop results, and to the smallest value of η0 for the open-loop results. The solid dot and the solid square indicate the intersection of the closed-loop Strehl-ratio kernel and the open-loop Strehl-ratio kernel for η0=1.4 and 1.0, respectively.

Fig. 9
Fig. 9

The optimal transition normalized spatial frequencies κ˜op and the expected Strehl ratio. The finite values of κ˜op as a function of η0 is plotted in Fig. 9a for the HV5/7 turbulence model. These results are used to obtain the (optimal) expected Strehl ratio (solid curve) shown in Fig. 9b for η0>0.824. For η00.824 the optimal expected Strehl ratio is calculated with the adaptive optics operating closed loop for all spatial frequencies. The dashed curve in Fig. 9b is the expected Strehl ratio calculated with the adaptive optics operating closed loop for κ˜<0.787 and operating open loop for κ˜>0.787. The Strehl ratio is calculated for a laser wavelength λ=1.0 µm.

Equations (102)

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UHEL(r, s, t)=U(r, s, t)exp[2πi(k¯s-ft)],
Ubeacon(r, s, t)=U(r, s, t)exp[-2πi(k¯s+ft)].
sU(r, s, t)- 1 2ik-12U(r, s, t)
-ikn(r, s, t)U(r, s, t)=0,
sU(r, s, t)+ 1 2ik-12U(r, s, t)
+ikn(r, s, t)U(r, s, t)=0.
n(r, s, t)=nT(r, s, t)+nH(r, s, t).
nT(r, s, t)=n¯T(r-v(s)t, s),
nH(r, s, t+Δt)=nH(r-v(s)Δt, s, t)+Δtμ(s)I(r, s, t),
tnH(r, s, t)+v(s)·nH(r, s, t)=μ(s)I(r, s, t).
U(r, s, t)=a(s, t)+dκ[bCR(κ, s, t)+ibCI(κ, s, t)]cos(2πκ·r)exp(-πiκ2λs)+dκ[bSR(κ, s, t)+ibSI(κ, s, t)]sin(2πκ·r)exp(-πiκ2λs),
U(r, s, t)=a(s, t)+dκ[bCR(κ, s, t)+ibCI(κ, s, t)]cos(2πκ·r)exp(πiκ2λs)+dκ[bSR(κ, s, t)+ibSI(κ, s, t)]sin(2πκ·r)exp(πiκ2λs),
nH(r, s, t)=dκ[νHC(κ, s, t)cos(2πκ·r)+νHS(κ, s, t)sin(2πκ·r)]r,
nT(r, s, t)=dκ[νTC(κ, s)cos(2πκ·v(s)t)-νTS(κ, s)sin(2πκ·v(s)t)]cos(2πκ·r)+dκ[νTC(κ, s)sin(2πκ·v(s)t)+νTS(κ, s)cos(2πκ·v(s)t)]sin(2πκ·r).
L0=μ0-10dsμ(s),
t0=Λ0/v0,v0=(CV2L02/3)1/2,
N˙λ=(λ)-10ds 1 2|a0|2μ(s)= 1 2|a0|2μ0L0/λ,
η0=N˙λt0.
zb˜(κ˜, z, τ)=A(κ˜, z)n˜H(κ˜, z, τ)+P(κ˜, V(z), z, τ)n˜T(κ˜, z),
zb˜(κ˜, z, τ)=A(κ˜, z)n˜H(κ˜, z, τ)+P(κ˜, V(z), z, τ)n˜T(κ˜, z),
τn˜H(κ˜, z, τ)=S(κ˜, V(z))n˜H(κ˜, z, τ)+4πη0α(z)Q(κ˜, z)b˜(κ˜, z, τ),
b˜(κ˜, z, τ)=b˜CR(κ˜, z, τ)b˜CI(κ˜, z, τ)b˜SR(κ˜, z, τ)b˜SI(κ˜, z, τ),
b˜(κ˜, z, τ)=b˜CR(κ˜, z, τ)b˜CI(κ˜, z, τ)b˜SR(κ˜, z, τ)b˜SI(κ˜, z, τ),
n˜H(κ˜, z, τ)=n˜HC(κ˜, z, τ)n˜HS(κ˜, z, τ),
n˜T(κ˜, z)=n˜TC(κ˜, z)n˜TS(κ˜, z),
A(κ˜, z)=-sin(πκ˜2z)0cos(πκ˜2z)00-sin(πκ˜2z)0cos(πκ˜2z),
A(κ˜, z)=-sin(πκ˜2z)0-cos(πκ˜2z)00-sin(πκ˜2z)0-cos(πκ˜2z),
S[κ˜, V(z)]=02πκ˜·V(z)-2πκ˜·V(z)0,
Q(κ˜, z)=cos(πκ˜2z)sin(πκ˜2z)0000cos(πκ˜2z)sin(πκ˜2z),
P(κ˜, V(z), z, τ)=-cos[2πκ˜·V(z)τ]sin[πκ˜2z]-sin[2πκ˜·V(z)τ]sin(πκ˜2z)cos[2πκ˜·V(z)τ]cos[πκ˜2z]sin[2πκ˜·V(z)τ]cos(πκ˜2z)-sin[2πκ˜·V(z)τ]sin[πκ˜2z]cos[2πκ˜·V(z)τ]sin(πκ˜2z)sin[2πκ˜·V(z)τ]cos[πκ˜2z]-cos[2πκ˜·V(z)τ]cos(πκ˜2z),
P(κ˜, V(z), z, τ)=-cos[2πκ˜·V(z)τ]sin(πκ˜2z)-sin[2πκ˜·V(z)τ]sin(πκ˜2z)cos[2πκ˜·V(z)τ]cos(πκ˜2z)sin[2πκ˜·V(z)τ]cos(πκ˜2z)-sin[2πκ˜·V(z)τ]sin(πκ˜2z)cos[2πκ˜·V(z)τ]sin(πκ˜2z)sin[2πκ˜·V(z)τ]cos(πκ˜2z)-cos[2πκ˜·V(z)τ]cos(πκ˜2z).
G(κ˜, z, τ; Z)R(κ˜, τ, V(Z))b˜(κ˜, z, τ),
G(κ˜, z, τ; Z)R(κ˜, τ, V(Z))b˜(κ˜, z, τ),
NH(κ˜, z, τ; Z)T(κ˜, τ, V(Z))n˜H(κ˜, z, τ),
zG(κ˜, z, τ; Z)=A(κ˜, z)NH(κ˜, z, τ; Z)+δ(z-Z)f(κ˜, z),
zG(κ˜, z, τ; Z)=A(κ˜, z)NH(κ˜, z, τ; Z)+δ(z-Z)f (κ˜, z),
τNH(κ˜, z, τ; Z)=S(κ˜, VΔ(z; Z))NH(κ˜, z, τ; Z)+4πη0α(z)Q(κ˜, z)G(κ˜, z, τ; Z),
VΔ(z; Z)=V(z)-V(Z),
f(κ˜, z)=-sin(πκ˜2z)cos(πκ˜2z)00,
f(κ˜, z)=-sin(πκ˜2z)-cos(πκ˜2z)00,
R(κ˜, τ, V(Z))=cos[2πκ˜·V(Z)τ]0sin[2πκ˜·V(Z)τ]00cos[2πκ˜·V(Z)τ]0sin[2πκ˜·V(Z)τ]-sin[2πκ˜·V(Z)τ]0cos[2πκ˜·V(Z)τ]00-sin[2πκ˜·V(Z)τ]0cos[2πκ˜·V(Z)τ],
T(κ˜, τ, V(Z))=cos[2πκ˜·V(Z)τ]sin[2πκ˜·V(Z)τ]-sin[2πκ˜·V(Z)τ]cos[2πκ˜·V(Z)τ].
NH(κ˜, z, 0; Z)=0.
G(κ˜, , τ; Z)=0.
G(κ˜, 0, τ; Z)=0openloop-BCLG(κ˜, 0, τ; Z)closeloop,
BCL=0000010000000001
P(τ)=|G(κ˜, , τ; Z)|={[GCR(κ˜, , τ; Z)]2+[GCI(κ˜, , τ; Z)]2+[GSR(κ˜, , τ; Z)]2+[GSI(κ˜, , τ; Z)]2}1/2,
σ(τ)={[GCI(κ˜, 0, τ; Z)]2+[GSI(κ˜, 0, τ; Z)]2}1/2.
zp±1/2=zp+zp±12,τq±1/2=τq+τq±12,
y(p± 1 2, q)=y(p, q)+y(p±1, q)2,
y(p, q±12)=y(p, q)+y(p, q±1)2.
G(p, q)-G(p-1, q)
=ΔzA(p-1 2)NH(p-1 2, q)+δp,Pf(p-1 2),
G(p, q)-G(p-1, q)
=ΔzA(p-1 2)NH(p-1 2, q)+δp,Pf(p-1 2),
NH(p-1 2, q)-NH(p-1 2, q-1)=Δτ{S[VΔ(p-1 2)]NH(p-1 2, q-1 2)+4πη0α(p-1 2)Q(p-1 2)G(p-1 2, q-1 2)}.
NH(κ˜, z, τ; Z)=4πη0α(z)0τdτcos[2πκ˜·VΔ(z; Z)(τ-τ)]sin[2πκ˜·VΔ(z; Z)(τ-τ)]-sin[2πκ˜·VΔ(z; Z)(τ-τ)]cos[2πκ˜·VΔ(z; Z)(τ-τ)]
×Q(κ˜, z)G(κ˜, z, τ; Z).
NH(p-1 2, q)=(Δz)-1zp-1zpdzNH(κ˜, z, τq; Z).
NH(p-1 2, q)=4πη0α(zp-1/2)0τqdτap(τq-τ)bp(τq-τ)-bp(τq-τ)ap(τq-τ)×Q(p-12)G(p-12, τ),
ap(τ)=(Np)-1k=0Np-1 cos(2πκ˜τVp,k+1/2)×sinc[πκ˜τ(Vp,k+1-Vp,k)],
bp(τ)=(Np)-1k=0Np-1 sin(2πκ˜τVp,k+1/2)×sinc[πκ˜τ(Vp,k+1-Vp,k)],
Z=0.5,κ˜=0.5,η0=0.5.
β(κ˜, z)b˜(κ˜, z, ),β(κ˜, z)b˜(κ˜, z, ),
ηT(κ˜, z)nT(κ˜, z),ηH(κ˜, z)n˜H(κ˜, z, ).
n˜H(κ˜, z, 0)=0.
n˜H(κ˜, z, τ)=4πη0α(z)0τdτcos[2πκ˜·V(z)(τ-τ)]-sin[2πκ˜·V(z)(τ-τ)]sin[2πκ˜·V(z)(τ-τ)]+cos[2πκ˜·V(z)(τ-τ)]×Q(κ˜, z)b˜(κ˜, z, τ).
ηH(κ˜, z)=2πη0α(z)(κ˜)-1Q(κ˜, z)β(κ˜, z).
sin[2πκ˜·V(z)τ]=0,
0dτcos[2πκ˜·V(z)τ]=(8π)-1/2(κ˜)-1.
n¯T(r, s)=dκ[νTC(κ, s)cos(2πκ·r)+νTS(κ, s)sin(2πκ·r)].
zβ(κ˜, z)=2πη0α(z)(κ˜)-1A(κ˜, z)Q(κ˜, z)β(κ˜, z)+P(κ˜, z)η˜T(κ˜, z),
zβ(κ˜, z)=2πη0α(z)(κ˜)-1A(κ˜, z)Q(κ˜, z)β(κ˜, z)+P(κ˜, z)η˜T(κ˜, z),
P(κ˜, z)=-sin(πκ˜2z)0cos(πκ˜2z)00sin(πκ˜2z)0-cos(πκ˜2z),
P(κ˜, z)=-sin(πκ˜2z)0-cos(πκ˜2z)00sin(πκ˜2z)0cos(πκ˜2z).
β(κ˜, 0)=0openloop-BCLβ(κ˜, 0)closeloop ,
β(κ˜, )=0.
zxR(κ˜, z; Z)=-π/2η0α(z)(κ˜)-1{sin(2πκ˜2z)xR(κ˜, z; Z)+[1-cos(2πκ˜2z)]xI(κ˜, z; Z)}-sin(πκ˜2z)δ(z-Z),
zxI(κ˜, z; Z)=π/2η0α(z)(κ˜)-1{[1+cos(2πκ˜2z)]xR(κ˜, z; Z)+sin(2πκ˜2z)xI(κ˜, z; Z)}+cos(πκ˜2z)δ(z-Z),
xR(κ˜, 0; Z)=0,xI(κ˜, 0; Z)=0,
zξR(κ˜, z)=-π/2η0α(z)(κ˜)-1{sin(2πκ˜2z)ξR(κ˜, z)+[1-cos(2πκ˜2z)]ξ1(κ˜, z)},
zξI(κ˜, z)=π/2η0α(z)(κ˜)-1{[1+cos(2πκ˜2z)]ξR(κ˜, z)+sin(2πκ˜2z)ξI(κ˜, z)},
ξR(κ˜, 0)=0,ξI(κ˜, 0)=1.
βCR(κ˜, z)=0dZηTC(κ˜, Z)PR(κ˜, z; Z),
βCI(κ˜, z)=0dZηTC(κ˜, Z)PI(κ˜, z; Z),
βSR(κ˜, z)=-0dZηTS(κ˜, Z)PR(κ˜, z; Z),
βSI(κ˜, z)=-0dZηTS(κ˜, Z)PI(κ˜, z; Z),
PR(κ˜, z; Z)
=xR(κ˜, z; Z)
+0,openloop-xI(κ˜, ; Z)ξR(κ˜, z)/ξI(κ˜, ),closedloop,
PI(κ˜, z; Z)
=xI(κ˜, z; Z)+0,openloop-xI(κ˜, ; Z)ξI(κ˜, z)/ξI(κ˜, ),closedloop.
zβCR(κ˜, z)=zβCR(κ˜, z),
zβCI(κ˜, z)=-zβCI(κ˜, z),
zβSR(κ˜, z)=zβSR(κ˜, z),
zβSI(κ˜, z)=-zβSI(κ˜, z).
ξI(κ˜, )=0.
SR=drW(r)UHEL(r, )2drW(r)×drW(r)|UHEL(r, )|2,
SR1+0.02288r0-5/3dκPAVG2(κ˜)κ-11/3-1,
PAVG2(κ˜)=0dsCN2(s)|PR(κ˜, ; s)+iPI(κ˜, ; s)|20dsCN2(s),
6.88r0-5/3=2.91k20dsCN2(s).
N˙λ=-(n0-1) L0λγ-1γp0μ0I0,

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