Sixth-order wave aberration theory of ultrawide-angle optical systems: erratum

Lijun Lu; Yiqing Cao

doi:10.1364/AO.57.001050

Abstract

In this erratum, two errors in our recently published paper [Appl. Opt. 56, 8570 (2017) [CrossRef] ] are corrected.

1. The term of $w_{i j 0 (1)}$ in Table 1 of Ref. [1] should be written as $A_{1}^{i} B_{1}^{j} w_{i j 0 (1)}$ ; thus we should define a new parameter ${\hat{w}}_{i j 0 (1)}$ to represent it, ${\hat{w}}_{i j 0 (1)} = A_{1}^{i} B_{1}^{j} w_{i j 0 (1)}$ In addition, the expression of $w_{240}^{(E)}$ in Table 1 is incorrect and needs to be revised. Consequently, Table 1 should be revised as follows: $w_{400}^{(E)} = - 9 ϕ_{1} {\hat{w}}_{300 (1)} w_{300 (2)} w_{220}^{(E)} = - 3 ϕ_{1} ({\hat{w}}_{120 (1)} w_{300 (2)} + {\hat{w}}_{300 (1)} w_{120 (2)}) + 4 ϕ_{2} {\hat{w}}_{120 (1)} w_{120 (2)} w_{040}^{(E)} = - ϕ_{1} {\hat{w}}_{120 (1)} w_{120 (2)} w_{500}^{(E)} = - 12 ϕ_{1} ({\hat{w}}_{300 (1)} w_{400 (2)} + {\hat{w}}_{400 (1)} w_{300 (2)}) w_{140}^{(E)} = - 2 ϕ_{1} ({\hat{w}}_{120 (1)} w_{220 (2)} + {\hat{w}}_{220 (1)} w_{120 (2)}) + 8 ϕ_{2} ({\hat{w}}_{120 (1)} w_{040 (2)} + {\hat{w}}_{040 (1)} w_{120 (2)}) w_{320}^{(E)} = - 2 ϕ_{1} [2 ({\hat{w}}_{120 (1)} w_{400 (2)} + {\hat{w}}_{400 (1)} w_{120 (2)}) + 3 ({\hat{w}}_{300 (1)} w_{220 (2)} + {\hat{w}}_{220 (1)} w_{300 (2)})] + 4 ϕ_{2} ({\hat{w}}_{120 (1)} w_{220 (2)} + {\hat{w}}_{220 (1)} w_{120 (2)}) w_{600}^{(E)} = - ϕ_{1} [15 ({\hat{w}}_{300 (1)} w_{500 (2)} + {\hat{w}}_{500 (1)} w_{300 (2)}) + 16 {\hat{w}}_{400 (1)} w_{400 (2)}] w_{240}^{(E)} = - ϕ_{1} [3 ({\hat{w}}_{120 (1)} w_{320 (2)} + {\hat{w}}_{320 (1)} w_{120 (2)} + {\hat{w}}_{140 (1)} w_{300 (2)} + {\hat{w}}_{300 (1)} w_{140 (2)}) + 4 {\hat{w}}_{220 (1)} w_{220 (2)}] + 8 ϕ_{2} ({\hat{w}}_{040 (1)} w_{220 (2)} + {\hat{w}}_{220 (1)} w_{040 (2)} + {\hat{w}}_{120 (1)} w_{140 (2)} + {\hat{w}}_{140 (1)} w_{120 (2)}) w_{420}^{(E)} = - ϕ_{1} [8 ({\hat{w}}_{220 (1)} w_{400 (2)} + {\hat{w}}_{400 (1)} w_{220 (2)}) + 9 ({\hat{w}}_{320 (1)} w_{300 (2)} + {\hat{w}}_{300 (1)} w_{320 (2)}) + 5 ({\hat{w}}_{500 (1)} w_{120 (2)} + {\hat{w}}_{120 (1)} w_{500 (2)})] + 4 ϕ_{2} ({\hat{w}}_{320 (1)} w_{120 (2)} + {\hat{w}}_{120 (1)} w_{320 (2)} + {\hat{w}}_{220 (1)} w_{220 (2)}) w_{060}^{(E)} = - ϕ_{1} ({\hat{w}}_{120 (1)} w_{140 (2)} + {\hat{w}}_{140 (1)} w_{120 (2)}) + 16 ϕ_{2} {\hat{w}}_{040 (1)} w_{040 (2)} .$ Accordingly, in the paragraph below Eq. (57), $w_{i j 0 (1)} \to {\tilde{w}}_{i j 0}$ should be revised to be ${\hat{w}}_{i j 0 (1)} \to {\tilde{w}}_{i j 0}$ .
2. The expression $c_{3, 2}^{*}$ is missing in Appendix A, and thus should have appeared as $c_{3, 2}^{*} = \frac{(c_{0, 2} c_{3, 0} + c_{1, 2} c_{2, 0}) \sin α \tan α}{r_{m}} - c_{3, 2} .$
The above two errors do not affect the numerical calculations in Section 5 in Ref. [1], since the calculations were based on the correct equations.

REFERENCE

1. L. Lu and Y. Cao, “Sixth-order wave aberration theory of ultrawide-angle optical systems,” Appl. Opt. 56, 8570–8583 (2017). [CrossRef]

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{\hat{w}}_{i j 0 (1)} = A_{1}^{i} B_{1}^{j} w_{i j 0 (1)}

w_{400}^{(E)} = - 9 ϕ_{1} {\hat{w}}_{300 (1)} w_{300 (2)} w_{220}^{(E)} = - 3 ϕ_{1} ({\hat{w}}_{120 (1)} w_{300 (2)} + {\hat{w}}_{300 (1)} w_{120 (2)}) + 4 ϕ_{2} {\hat{w}}_{120 (1)} w_{120 (2)} w_{040}^{(E)} = - ϕ_{1} {\hat{w}}_{120 (1)} w_{120 (2)} w_{500}^{(E)} = - 12 ϕ_{1} ({\hat{w}}_{300 (1)} w_{400 (2)} + {\hat{w}}_{400 (1)} w_{300 (2)}) w_{140}^{(E)} = - 2 ϕ_{1} ({\hat{w}}_{120 (1)} w_{220 (2)} + {\hat{w}}_{220 (1)} w_{120 (2)}) + 8 ϕ_{2} ({\hat{w}}_{120 (1)} w_{040 (2)} + {\hat{w}}_{040 (1)} w_{120 (2)}) w_{320}^{(E)} = - 2 ϕ_{1} [2 ({\hat{w}}_{120 (1)} w_{400 (2)} + {\hat{w}}_{400 (1)} w_{120 (2)}) + 3 ({\hat{w}}_{300 (1)} w_{220 (2)} + {\hat{w}}_{220 (1)} w_{300 (2)})] + 4 ϕ_{2} ({\hat{w}}_{120 (1)} w_{220 (2)} + {\hat{w}}_{220 (1)} w_{120 (2)}) w_{600}^{(E)} = - ϕ_{1} [15 ({\hat{w}}_{300 (1)} w_{500 (2)} + {\hat{w}}_{500 (1)} w_{300 (2)}) + 16 {\hat{w}}_{400 (1)} w_{400 (2)}] w_{240}^{(E)} = - ϕ_{1} [3 ({\hat{w}}_{120 (1)} w_{320 (2)} + {\hat{w}}_{320 (1)} w_{120 (2)} + {\hat{w}}_{140 (1)} w_{300 (2)} + {\hat{w}}_{300 (1)} w_{140 (2)}) + 4 {\hat{w}}_{220 (1)} w_{220 (2)}] + 8 ϕ_{2} ({\hat{w}}_{040 (1)} w_{220 (2)} + {\hat{w}}_{220 (1)} w_{040 (2)} + {\hat{w}}_{120 (1)} w_{140 (2)} + {\hat{w}}_{140 (1)} w_{120 (2)}) w_{420}^{(E)} = - ϕ_{1} [8 ({\hat{w}}_{220 (1)} w_{400 (2)} + {\hat{w}}_{400 (1)} w_{220 (2)}) + 9 ({\hat{w}}_{320 (1)} w_{300 (2)} + {\hat{w}}_{300 (1)} w_{320 (2)}) + 5 ({\hat{w}}_{500 (1)} w_{120 (2)} + {\hat{w}}_{120 (1)} w_{500 (2)})] + 4 ϕ_{2} ({\hat{w}}_{320 (1)} w_{120 (2)} + {\hat{w}}_{120 (1)} w_{320 (2)} + {\hat{w}}_{220 (1)} w_{220 (2)}) w_{060}^{(E)} = - ϕ_{1} ({\hat{w}}_{120 (1)} w_{140 (2)} + {\hat{w}}_{140 (1)} w_{120 (2)}) + 16 ϕ_{2} {\hat{w}}_{040 (1)} w_{040 (2)} .

c_{3, 2}^{*} = \frac{(c_{0, 2} c_{3, 0} + c_{1, 2} c_{2, 0}) \sin α \tan α}{r_{m}} - c_{3, 2} .

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Applied Optics