Seventh-order aberrations of gradient-index media with axial symmetry

Leon Magiera

doi:10.1364/JOSAA.12.000794

Journal of the Optical Society of America A
Vol. 12,
Issue 4,
pp. 794-800
(1995)
•https://doi.org/10.1364/JOSAA.12.000794

Seventh-order aberrations of gradient-index media with axial symmetry

Leon Magiera

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Leon Magiera, "Seventh-order aberrations of gradient-index media with axial symmetry," J. Opt. Soc. Am. A 12, 794-800 (1995)

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Abstract

The theory of the seventh-order aberrations of gradient-index media with axial symmetry is presented. This theory is based on Hamilton’s equations. Formulas for seventh-order aberrations are derived. On the basis of the derived formulas the explicit algebraic expressions for all seventh-order aberration coefficients for gradient-index media with cylindrical symmetry are evaluated. The correctness of the obtained coefficients is verified for secant hyperbolic exact and helical exact gradient-index distributions. It is also shown that, for meridional rays, some of the aberration coefficients vanish for other index distributions besides the secant hyperbolic exact gradient-index distribution.

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Equations (61)

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Equation	Parameter Values

	β	γ	δ
τ₀ = 0	$\frac{5}{12}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₁₀ = 0	$\frac{5}{12}$	$- \frac{61}{720}$	Arbitrary
τ₁₀ = 0	$\frac{\frac{161}{4} m + \frac{175}{6}}{m - 10}$	$- \frac{61}{720}$	Arbitrary
τ₂₀ = 0	$\frac{5}{12}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₂₀ = 0	$\pm \frac{[{(5854680 m^{2} π^{2} - 9457560 m π^{2} + 2814750 π^{2} + 7348240)}^{1 / 2} + 195 m^{2} π^{2} - 315 m π^{2} + \frac{375}{4} π^{2} + \frac{271}{2}]}{15 (468 m^{2} π^{2} - 756 m π^{2} + 225 π^{2} - 166)}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₃₀ = 0	$\frac{5}{12}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₃₀ = 0	$\frac{29 m - \frac{95}{4}}{12 m - 1}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₄₀ = 0	$\frac{5}{12}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₅₀ = 0	$\frac{5}{12}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₅₀ = 0	$\frac{\frac{145}{12} m - \frac{157}{12}}{5 m - 1}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₆₀ = 0	$\frac{5}{12}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₆₀ = 0	$\frac{5}{12} \pm \frac{[\frac{1}{5} {(- 4200 m^{2} π^{2} + 17640 m π^{2} - 6930 π^{2} + 129360)}^{1 / 2} + 25 m^{2} π^{2} - 105 m π^{2} + \frac{165}{4} π^{2} + \frac{175}{6}]}{60 m^{2} π^{2} - 252 m π^{2} + 99 π^{2} - 98}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₇₀ = 0	$\frac{5}{12}$	$- \frac{61}{720}$	$\frac{277}{8064}$
τ₇₀ = 0	$- \frac{169}{36}$	$- \frac{61}{720}$	$\frac{277}{8064}$

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Equations (61)

Journal of the Optical Society of America A