Abstract

It is shown that when branch points are present in the phase of a turbulence-distorted optical field, the ability of an adaptive optics system that utilizes a least mean square error type of wave-front reconstructor to sense all of the turbulence-induced phase perturbations is limited. There is a portion of the turbulence-induced phase perturbation, which portion we refer to as the hidden phase, that such a least mean square error type of wave-front reconstructor will, in effect, ignore. It is shown that the presence of branch points indicates that the measured phase-difference vector field cannot be considered to be simply the gradient of some scalar potential—the phase function—but is in part also the curl of a vector potential function. A solution is developed for this vector potential, and from this a simple closed-form solution for the hidden phase is developed. Sample numerical results are presented showing the nature of the hidden phase. Suggestions are provided for a branch-point-tolerant wave-front reconstructor based on use of the closed-form solution for the hidden phase.

© 1998 Optical Society of America

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References

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  1. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  2. V. Aksenov, V. Banakh, O. Tikhomirova, “Potential and vortex features of optical speckle fields and visualization of wave-front singularities,” Appl. Opt. 37, 4536–4540 (1998).
    [CrossRef]
  3. F. S. Woods, Advanced Calculus (Ginn, Boston, 1934), Eq. (80.8).
  4. W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Eq. (1-1).
  5. R. V. Churchill, Introduction to Complex Variables and Applications (McGraw-Hill, New York, 1948), Sec. 18.

1998

1992

Appl. Opt.

Other

F. S. Woods, Advanced Calculus (Ginn, Boston, 1934), Eq. (80.8).

W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Eq. (1-1).

R. V. Churchill, Introduction to Complex Variables and Applications (McGraw-Hill, New York, 1948), Sec. 18.

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Figures (2)

Fig. 1
Fig. 1

Hidden phase ϕhid for a single pair of branch points. A pair of branch points are located on the diagonal of the square region shown, one positive and the other negative. The contour lines indicate loci of constant value for the hidden phase ϕhid for values equal to 2π[-0.9,-0.8,,+0.9]. The contour line corresponding to ϕhid=0 is the dotted line along the diagonal. The results were calculated with Eq. (32).

Fig. 2
Fig. 2

Hidden phase ϕhid for a turbulence-distorted optical field. Plane-wave-propagation simulations were carried out for a strength of turbulence corresponding to a Rytov number of σl2=0.5 by use of a 256×256 array with 20 turbulence phase screens. The phase screens were generated by fast-Fourier-transform processing a suitably weighted array of random numbers, the weights being such that the phase screens nominally had a power spectral density corresponding to Kolmogorov turbulence. No additional tip-tilt or other lower-order terms were added, so the phase screens considered as a periodic pattern (period 256×256) nominally had no discontinuity at the period boundaries. The plane wave filled the full 256×256 array where it was launched, so the results are just as meaningful at the edges of the array as anywhere else on the array. (a) Logarithm of the resultant intensity pattern. (b) Branch points of the optical field: positive branch points are shown as white dots, the negative branch points as black dots. [The branch points were identified by calculating the circulation of g(rp,q), cf. Eq. (4), around each unit square of the computational array and considering there to be a branch point in the center of each square for which the value of the circulation was ±2π: a positive or a negative branch point according to whether the value was +2π or -2π. All the circulation values that were not equal to ±2π were found to be equal to zero to within computational accuracy.] (c) Contours of the hidden phase ϕhid calculated in accordance with Eq. (32) with use of the branch point location type of information of (b); contour lines are shown for ϕhid=2π[-0.9,-0.8,, 0.9]. This is the part of the turbulence-induced phase perturbation that an adaptive optics system with a least mean square error wave front reconstructor will not be able to see.

Equations (52)

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φ(r)=tan-1Im{u(r)}Re{u(r)}
φ(rp,q)=tan-1Im{u(rp,q)}Re{u(rp,q)},
g(rp,q)=PV{φ(rp+1,q)-φ(rp,q)}d1X+PV{φ(rp,q+1)-φ(rp,q)}d1Y,
g(r)=limδ0PV{φ(r+1Xδ)-φ(r)}δ1X+limδ0PV{φ(r+1Yδ)-φ(r)}δ1Y.
g(rp,q)
=d-1tan-1Im{u[(p+1)d,qd]u*(pd, qd)}Re{u[(p+1)d,qd]u*(pd, qd)}1X+d-1tan-1Im{u[pd,(q+1)d]u*(pd, qd)}Re{u[pd,(q+1)d]u*(pd, qd)}1Y,
g(r)=limδ0δ-1 tan-1Im{u(r+1Xδ)u*(r)}Re{u(r+1Xδ)u*(r)}1X+limδ0δ-1 tan-1Im{u(r+1Yδ)u*(r)}Re{u(r+1Yδ)u*(r)}1Y.
g(rp,q)·1Xd+g(rp+1,q)·1Yd-g(rp,q+1)·1X×d-g(rp,q)·1Yd
=±2πifabranchpointisenclosed0ifnobranchpointisenclosed.
Cdξt(ξ)·g(r(ξ))
=±2πifabranchpointisenclosed0ifnobranchpointisenclosed.
ϕlmse=Rg.
R=(ΓTΓ)-1ΓT,
Cdr1z·×g(r)
=±2πifabranchpointisenclosed0ifnobranchpointisenclosed.
1z·×g(r)=±2πδ(r-rbp).
g(r)=s(r)+×H(r).
H(r)=[0, 0, h(r)],
1z·××H(r)=±2πδ(r-rbp).
××H(r)=-2H(r)+·H(r).
·H(r)=0.
1z·××H(r)=-2h(r).
2h(r)=2πδ(r-rbp).
h(r)=log(|r-rbp|).
B=ΓA,
A=ΓTB,
A(rp,q)=d-1[BX(rp-1,q)-BX(rp,q)+BY(rp,q-1)-BY(rp,q)].
2ϕlmse(rp,q)=2s(rp,q),
2ϕlmse(rp,q)=·g(rp,q),
ϕ(r)=ϕlmse(r)+ϕhid(r).
g(r)=ϕ(r)=ϕlmse(r)+ϕhid(r).
ϕlmse(r)=s(r).
ϕhid(r)=×H(r),
ϕhid(r)x=h(r)y,ϕhid(r)y=-h(r)x.
ϕhid(r)x=-[-h(r)]y,ϕhid(r)y=[-h(r)]x,
ϕhid(r)=Im{±log[(x-xbp)+i(y-ybp)]}.
ϕhid(r)=Imlogk=1K(x-xk)+i(y-yk)k=1K(x-xk)+i(y-yk)>.
2s(r)=·g(r).
g(r)=ϕ(r).
ϕ(r)=ϕlmse(r)+ϕhid(r),
2ϕlmse(r)=·g(r).
h(r)=log(r),
2h(r)=12 log(r)r2+1r log(r)r+1r22 log(r)θ2+2 log(r)z2=12 log(r)r2+1r log(r)r!=1-r-2+1rr-1=0ifr0indeterminateifr=0·
Cdξt(ξ)·{s(r(ξ))+×H(r(ξ))}=±2π,
02πrdθ1θ·1r s(r)r+1θ 1rs(r)θ+1z s(r)z
+1r 1r[1z·H(r)]r-r [10·H(r)]z
+1θ[1r·H(r)]z-[1z·H(r)]r
+1z 1rr [1θ·H(r)]r-[1r·H(r)]θ
=02πrdθ1rs(r)θ+[1r·H(r)]z-[1z·H(r)]r=±2π.
02πrdθ[1r·H(r)]z-[1z·H(r)]r=±2π.
02πrdθ h(r)r=2π.
102πrdθr-1=2π,

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