The vortex structure and its field dependence are studied in the clean limit on the basis of the quasi-classical Eilenberger theory to find their difference between \documentclass[12pt]{minimal}
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$$d_{x^2 - y^2 }$$
\end{document}- and s-wave pairings. We show the \documentclass[12pt]{minimal}
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$$d_{x^2 - y^2 }$$
\end{document}-wave nature and the vortex lattice effect on the local density of states around the vortex, and on the field dependence of the spatially averaged density of states. The \documentclass[12pt]{minimal}
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$$d_{x^2 - y^2 }$$
\end{document}-wave pairing introduces a fourfold symmetric structure around each vortex core in the pair potential and the internal field. With increasing field, their contribution becomes significant to the whole structure of the vortex lattice state.
