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Figure 1: The four different flexion fields discussed in the text. The upper left (right) panel shows the flexion corresponding to an axially-symmetric E-mode (B-mode) shear field, where arrows indicate the spin-1 flexion and the skeletons the spin-3 flexion component. In the lower left (right) panel, the flexion fields are displayed which are not due to a shear field, but a non-zero E-mode (B-mode) field.

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In the text

Figure 2: Constraints on the combination of source size and reduced flexion for the validity of the concept of flexion. Each curve shows the dividing line between a circular source of limiting isophote being cut by a caustic (above the curve) or not (below the curve); in the former case, the assumptions underlying the flexion concept break down. The different curves in each panel are for different values of g, chosen as g=0.4,0.2,0.1,0.05,0, as indicated by different line types. Without loss of generality, we choose g to be real and non-negative. The four panels differ in the phase of the reduced flexion, as indicated. E.g., in the upper left panel, the phases of G1, G3 are the same as that of g.

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In the text

Figure 3: Accuracy of the estimates for reduced shear and flexion. The left panel shows contour of constant fractional error of , and , on the estimate of the reduced shear g, as a function of , where we chose g=0.05 as input value, and assumed the phases of G1, G3 to be the same as that of g. The estimate was obtained by solving the iteration equations given in Sect. 5. The right panel shows the fractional error levels at 3, 5, and 10% for the reduced flexion, as quantified by (55), where the estimate was obtained again with the iterative procedure. In both cases, we assumed circular sources.

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In the text

Figure 4: Comparison of the reduced flexion estimators (53) with the full expression (46) and the input value. The horizontal and vertical axis show . For both panels, we take g=0.05, and G3=0 (G1=0) for the left (right) panel. The line indicates the input value, the plus symbols show the simplified reduced flexion estimate (53), and the crosses result from the full expression of reduced flexion (46). As can be seen from the left-hand panel, the full estimator for the reduced flexion yields a more biased result that the approximate expression (53); we have not found a reasonable explanation for this behavior.

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In the text

Figure A.1: The critical curves ( left-hand panel) and caustics ( right-hand panel) of the lens Eq. (9) for the cases of hyperbolic critical curves, as described in Sect. B.2. The parameters chosen here are g=0.05, , . A circular source is mapped onto two images, as indicated. If the source size were increased, it would hit the caustic, the two images would merge, and the flexion concept would break down. The unit of the reduced flexion is the inverse of the unit in which coordinates are measured.

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In the text

	Figure A.2: Same as Fig. A.1, but for the parabolic case, with parameters g=0.05, G1=-0.04, G3=0.112.
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In the text

	Figure A.3: Same as Fig. A.1, but for the elliptical case, with parameters .
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In the text