Abstract

It is shown that, when the scalar field associated with the propagation of a distorted wave function has nulls in its intensity pattern, the phase function that goes with that scalar field has branch points at the location of these nulls and that there are unavoidable 2π discontinuities across the associated branch cuts in the phase function. An analytic proof of this supposition is provided. Sample computer-wave optics propagation results are presented that manifest such unavoidable discontinuities. Among other things, the numerical results are organized in a way that demonstrates that for those cases the branch points are unavoidable. It is found in the sample numerical results that the branch cuts can be positioned so that the 2π discontinuities are located along lines of minimum intensity. This location tends to minimize the physical significance or importance of the discontinuities, a significant consideration for deformable-mirror adaptive optics, for which there is an unavoidable correction error in the vicinity of the branch cut. An algorithm is briefly described that allows the branch cuts to be located automatically and a phase function to be calculated that has discontinuities equal only to 2π discontinuities that are located at the branch cuts.

© 1992 Optical Society of America

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References

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  1. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  2. H. Takajo, T. Takahashi, “Noniterative method for obtaining the exact solution for the normal equation in least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
    [CrossRef]
  3. M. J. Northcott, G. R. Ayers, J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
    [CrossRef]

1988 (3)

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Figures (18)

Fig. 1
Fig. 1

Closed circle contour in (x, y) space centered at (x0, y0). The closed contour is a circle in (x, y) space having radius ρ with its center at (x0, y0). The angle θ may be considered to be the parameter specifying the position of a point on the contour. As θ varies in value from θ = 0 to θ = 2π, the point moves around the circular contour in a positive, i.e, counterclockwise, fashion.

Fig. 2
Fig. 2

Mapping of the circle in (x, y) space onto (u, v) space with u0 and v0 not equal to zero. The mapping of the circular contour in (x, y) space that is shown in Fig. 1, onto (u, v) space, as shown here, is carried out in accordance with Eqs. (10) and (11) or in accordance with approximations (20) and (21). The circle shown in Fig. 1 goes into the ellipse shown here. The values of the major and minor axes of this ellipse, 2a and 2b, are defined by Eqs. (28) and (29), whereas the value of the angle Θ between these axes and the (u, v) axes is defined by Eq. (30). Because u0 and v0 are both nonzero it is possible to pick a value of the radius ρ of the circle small enough that the ellipse is entirely contained in a single quadrant. (In preparing this drawing we assumed that u0 and v0 are both positive, so the ellipse appears in the first quadrant. Other cases with the ellipse appearing in a different quadrant are equally compatible with our presentation.) The mapping yields a contour that we go about in a positive sense (i.e., counterclockwise) just as we did for the circular contour in Fig. 1. However, the mapping could just as well have resulted in a reversal of the handedness of the contour, a curve that we would go around in the opposite sense, i.e., negatively or clockwise. Without loss of generality we may (and shall) restrict our attention here and in the following figures to consideration of a positive handedness mapping but will recall that consideration of a negative handedness mapping is just as reasonable. The phase φ(x, y), defined by definitions (7)–(9), is equal to the angle between the u axis and the radius vector from the origin in (u, v) space to the point on the contour, at u(x, y), v(x, y), that the point (x, y) maps into in (u, v) space.

Fig. 3
Fig. 3

Mapping of the circle in (x, y) space onto (u, v) space with u0 equal to zero and positive v0. In this case, unlike for that of Fig. 2, the mapping of the circular contour shown in Fig. 1 onto (u, v) space in accordance with approximations (20) and (21) results in an elliptical contour split between the first and second quadrants, which could have been between the third and fourth quadrants if we had a negative value for v0 instead of the positive value assumed here. Although the mapping is positive in its handedness, we understand that essentially the same argument that is developed in the text would apply if we had illustrated a case in which the mapping was negative in its handedness.

Fig. 4
Fig. 4

Mapping of the circle in (x, y) space onto (u, v) space with v0 equal to zero and negative u0. Because u0 is negative and v0 is equal to zero, the contour appears split between the second and the third quadrants.

Fig. 5
Fig. 5

Mapping onto (u, v) space with both u0 and v0 equal to zero. Here, the circular contour shown in Fig. 1 maps into the elliptical contour centered at the origin, as shown here. In this case a part of the mapped contour lies in each of the four quadrants. Because both u0 and v0 are equal to zero, then no matter how small we choose the radius ρ of the circular contour in Fig. 1 to be, parts of the circular contour will necessarily map into each of the four quadrants. This mapping is unavoidable and is central to the fact for this case that there is a 2π phase discontinuity necessarily encountered in going around the contour. This branch point property cannot be avoided when u0 and v0 are both equal to zero, i.e., when the corresponding optical intensity at (x0, y0) is zero. Although our illustration indicates a mapping with a positive treatment of handedness so that a positive loop (counterclockwise) in Fig. 1 goes into a positive loop here, it is equally possible for the mapping to have a negative treatment of handedness, in which case a positive loop in Fig. 1 would go into a negative loop (clockwise) here. One treatment of handedness by the mapping will result in a +2π phase discontinuity encountered as we go around the contour, whereas the other treatment of handedness will result in a −2π phase discontinuity encountered as we go around the contour.

Fig. 6
Fig. 6

Randomly distorted phase screen used in focused beam propagation calculations. The phase screen used to distort the wave front, which is otherwise focused and whose intensity at the focal pane is shown in Fig. 7, is depicted here. The propagation calculation was performed on a 512 × 512 grid. Here, we show the phase for the central 256 × 256 portion of the grid.

Fig. 7
Fig. 7

Intensity at focus for a randomly distorted circular wave front; the darker the region, the higher the intensity. The wave-optics propagation calculations used to generate these results used a 512 × 512 computational grid corresponding to an L × L area, where L/D = 2 and D denotes the diameter of the laser transmitter. The random wave-front distortion is introduced as a single turbulent phase screen immediately in front of the aperture, with an r0 such that D/r0 = 10. The range R at which the laser beam is nominally in focus and that corresponds to the intensity pattern shown here is such that L/(Rλ/D) = 20.48, λ denoting the laser wavelength. We show the central (L/2) × (L/2) portion of the region calculated, i.e., only the central 256 × 256 points in the computational grid.

Fig. 8
Fig. 8

Probability distribution of the difference of phases at adjacent grid points for focused beam propagation. The propagation case is the one that gave rise to the intensity pattern shown in Fig. 7. The phase differences are calculated in accordance with Eqs. (39) and (40).

Fig. 9
Fig. 9

Location of branch points for a focused beam. The propagation case considered is the same as that for which the intensity pattern is shown in Fig. 7. The scale of this figure is essentially the same as that for Fig. 7, and it is appropriate to overlay one figure upon the other for comparison purposes. Shown here is the sum around the elemental squares of the computational grid of the phase differences computed according to Eqs. (39) and (40). The sums are found to have only three possible values; most are nearly equal to zero, indicated by gray dots. A few are found to have values near 2π, indicated by black dots. We associate these with a positive-sense branch point in the phase function ϕ(x, y). A comparable number of sums are found to have values near −2π wr and are indicated here by white dots. We associate these with a negative-sense branch point in the phase function. Positive- and negative-sense branch points generally appear to occur in pairs, but it is not always obvious exactly how the pairings should be formed. (The pairings would, of course, define the two ends of a branch cut.) It is apparent from a comparison of Fig. 7 with this figure that the branch points appear to occur at points of near-zero intensity and that often a trough of relatively low intensity connects an obvious pairing of branch points.

Fig. 10
Fig. 10

Contour of the zeroes of u(x, y) and v(x, y) for the focused beam. This propagation case is the one for which the intensity pattern is shown in Fig. 7. The solid curves indicate values of (x, y) for which u(x, y) = 0, the dashed curves show values of (x, y) for which v(x, y) = 0. Where the solid and the dashed curves cross, the intensity is strictly equal to zero. A computer analysis of these contours showed that there was a crossing of a solid and a dashed curve in every elemental square of our grid for which a ±2π branch point, as shown in Fig. 9, was found to occur.

Fig. 11
Fig. 11

Effect of computational resolution on the apparent occurrence of branch points for a focused beam. The results correspond to those of Fig. 9, except that the number of sample points in the computational grid has been varied, with the overall physical size of the grid the same in all cases. Here we show both the positive and the negative branch points by black squares. The computational grid sizes were (a) 64 × 64, with only the central 32 × 32 being shown; (b) 128 × 128, with only 64 × 64 shown; (c) 256 × 256, with only 128 × 128 shown; and (d) 512 × 512 (the same as for Fig. 9), with only 256 × 256 shown. The number of branch points appears to increase by approximately a factor of 1.5× each time we double the computational resolution. The new branch points appear to show up in pairs (one positive and one negative), which were combined in one elemental square at lower resolution and therefore canceled each other out at the lower computational resolution. Apparently, at higher computational resolution the positive and negative branch points are resolved, and their individual existence becomes apparent.

Fig. 12
Fig. 12

Phase function, φ(x, y) with 2π discontinuities for a focused aberrated laser beam. The results correspond to the same propagation case of a focused aberrated laser beam that gave rise to the results shown in Figs. 7 and 9. The phase values were generated by using an intensity weighted least-squares reconstruction from the phase differences produced by application of Eqs. (39) and (40) to yield these calculated scalar field results. There are clearly defined 2π discontinuities apparent. (That they are discontinuities is obvious. That their size is 2π can be inferred only from consideration of Figs. 15 and 16.) Figure 13 shows the location of the approximately 2π-size phase differences, which we associate with branch cuts in the phase function. The image shown here (and in Figs. 13 and 14) corresponds to the central 256 × 256 portion of the 512 × 512 computational grid.

Fig. 13
Fig. 13

Branch cuts for a focused aberrated laser beam. The phase function shown in Fig. 12 has been differenced, and all differences not close to ±2π in magnitude have been set to zero to generate the results shown here. The 2π size differences are identified with branch cuts in the phase function. Note, in comparing this figure with Fig. 7, that the branch cuts shown here all tend to fall in regions of low intensity; note, in comparing this figure with Fig. 9, that the branch cuts end on the branch points. This image corresponds to the central 256 × 256 portion of the 512 × 512 computational grid.

Fig. 14
Fig. 14

Overlay of Figs. 7 and 13 of the intensity pattern and the branch cuts for the focused aberrated beam case. Obviously, our phase construction scheme has succeeded in tucking away the phase discontinuities (i.e., the branch cuts) in regions of low intensity.

Fig. 15
Fig. 15

Cumulative probability distribution of phase-difference values for a focused aberrated laser beam. This set of phase-difference values corresponds to the difference of phase values at adjacent grid points for the phase function shown in Fig. 12.

Fig. 16
Fig. 16

Cumulative weighted probability distribution of phase-difference values for a focused aberrated laser beam. This material is identical to that shown in Fig. 15, except that, in forming the cumulative probability distribution for Fig. 15 we assigned each phase difference was a weight of unity; in calculating the cumulative weighted distribution shown here, we assigned each phase difference a weight corresponding to the estimated minimum intensity on the line between the two points for which the phase difference is formed.

Fig. 17
Fig. 17

Intensity and branch cuts overlay for collimated beam propagation. The results are the same as in Fig. 14 but for the propagation of a collimated beam. On transmission the collimated beam is passed through a turbulencelike wave-front distortion screen, the strength of which is characterized by the same r0 value as for the screen used in developing the focused beam results shown in Fig. 14. Results for collimated beam propagation out to a range R = αr02/λ, where a = 0.1, 0.3, 0.5, 0.7, 1.0, 1.5, 2.0, 3.0, and 4.0, are shown in (a), (b), (c), (d), (e), (f), (g), (h), and (i), respectively.

Fig. 18
Fig. 18

Phase for collimated beam propagation. These results are the same as in Fig. 12, but for the propagation of a collimated beam. These phase results correspond to the same propagation runs as for the results shown in Fig. 17: (a)–(i) correspond to the respective parts of Fig. 17.

Equations (42)

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ϕ ( r ) = Arg { U ( r ) } + 2 π n ( r ) ,
n ( r ) = , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 ,
U ( r ) = U ( r ) exp [ i ϕ ( r ) ] ,
r ( x , y )
U ( x , y ) = u ( x , y ) + i v ( x , y ) = ( u 0 + i v 0 ) + ( u x + i v x ) ( x - x 0 ) + ( u y + i v y ) ( y - y 0 ) + ,
ϕ ( x , y ) = 2 π n ( x , y ) + φ ( x , y ) ,
- π < φ ( x , y ) π ,
cos [ φ ( x , y ) ] = u ( x , y ) / { [ u ( x , y ) ] 2 + [ v ( x , y ) ] 2 } 1 / 2 ,
sin [ φ ( x , y ) ] = v ( x , y ) / { [ u ( x , y ) ] 2 + [ v ( x , y ) ] 2 } 1 / 2 ,
u ( x , y ) = u 0 + u x ( x - x 0 ) + u y ( y - y 0 ) + ,
v ( x , y ) = v 0 + v x ( x - x 0 ) + v y ( y - y 0 ) + .
x = x 0 + ρ cos θ ,
y = y 0 + ρ sin θ .
u ( x , y ) = u 0 + u x ρ cos θ + u y ρ sin θ + ,
v ( x , y ) = v 0 + v x ρ cos θ + v y ρ sin θ + .
μ u = [ ( u x ρ ) 2 + ( u y ρ ) 2 ] 1 / 2 ,
μ v = [ ( v x ρ ) 2 + ( v y ρ ) 2 ] 1 / 2 ,
θ u = arctan 2 ( u x , u y ) ,
θ v = arctan 2 ( v x , v y ) ,
u ( x , y ) = u 0 + ( μ u cos θ u ) cos θ + ( μ u sin θ u ) sin θ + u 0 + μ u cos ( θ - θ u ) ,
v ( x , y ) = v 0 + ( μ v cos θ v ) cos θ + ( μ v sin θ v ) sin θ + v 0 + μ v cos ( θ - θ v ) .
A = μ u 2 cos 2 θ u + μ v 2 cos 2 θ v ,
B = 2 μ u 2 cos θ u sin θ u + 2 μ v 2 cos θ v sin θ v ,
C = μ u 2 sin 2 θ u + μ v 2 sin 2 θ v ,
R 2 = ( u - u 0 ) 2 + ( v - v 0 ) 2 = A cos 2 θ + B cos θ sin θ + C sin 2 θ ,
( R 2 θ ) θ = ϑ = 0.
ϑ = 1 / 2 tan - 1 ( B / A - C ) .
a 2 = A cos 2 ϑ + B cos ϑ sin ϑ + C sin 2 ϑ ,
b 2 = A cos 2 ( ϑ + 1 / 2 π ) + B cos ( ϑ + 1 / 2 π ) × sin ( ϑ + 1 / 2 π ) + C sin 2 ( ϑ + 1 / 2 π ) = A sin 2 ϑ - B cos ϑ sin ϑ + C cos 2 ϑ .
Θ = tan - 1 [ μ v cos ( ϑ - θ v ) μ u cos ( ϑ - θ u ) ] .
u = u 0 + u cos Θ - v sin Θ ,
v = v 0 + u sin Θ + v cos Θ ,
u = a cos ( θ - ϑ ) ,
v = b sin ( θ - ϑ ) .
I ( x , y ) = ½ u ( x , y ) + i v ( x , y ) 2 = ½ { [ u ( x , y ) ] 2 + [ v ( x , y ) ] 2 } ,
( x 0 , y 0 ) d θ ϕ θ ( ρ cos θ , ρ sin θ ) = { ± 2 π , if u 0 = 0 , v 0 = 0 0 otherwise .
d θ ϕ θ ( ρ cos θ , ρ sin θ ) = { ± 2 π , if I ( x 0 , y 0 ) = 0 0 if I ( x 0 , y 0 ) > 0 .
[ ϕ ( r ) - ϕ ( r ) ] 2 = 6.88 ( r - r / r 0 ) 5 / 3 .
Δ x φ ( x , y ) = φ ( x + Δ , y ) - φ ( x , y ) + 2 π n x ( x , y ) ,
Δ y φ ( x , y ) = φ ( x , y + Δ ) - φ ( x , y ) + 2 π n y ( x , y ) ,
exp [ i ϕ ( x , y ) ] = U ( x , y ) / U ( x , y ) ,
R = α r 0 2 / λ .

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