The purpose of this note is to study the three manifold invariants Qp(ML).For lens spaces L(7,1)and (7,2),we compute Q5(L(7,1) and Q5(L(7.2)) concretely,which enables us to prove that the in-variants can distinguish h...The purpose of this note is to study the three manifold invariants Qp(ML).For lens spaces L(7,1)and (7,2),we compute Q5(L(7,1) and Q5(L(7.2)) concretely,which enables us to prove that the in-variants can distinguish homotopy equivalence manifolds.展开更多
An effective design method of freeform micro lens array is presented for shaping varied laser beams into prescribed rectangular illumination. The variable separation mapping is applied to design concave freeform surfa...An effective design method of freeform micro lens array is presented for shaping varied laser beams into prescribed rectangular illumination. The variable separation mapping is applied to design concave freeform surfaces for constructing a freeform lens array. Several dedicated examples show that the designed freeform optical lens array can achieve a prescribed rectangular illumination pattern, especially without considering the initial states of incident laser beams. Both high collection efficiency and good spatial uniformity can be available simultaneously. Tolerance analysis is also performed to demonstrate that this optical device can well avoid fabricating difficulty in actual applications.展开更多
文摘The purpose of this note is to study the three manifold invariants Qp(ML).For lens spaces L(7,1)and (7,2),we compute Q5(L(7,1) and Q5(L(7.2)) concretely,which enables us to prove that the in-variants can distinguish homotopy equivalence manifolds.
基金supported by the National Natural Science Foundation of China(No.61405037)the Science and Technology Project of Fujian Province(No.2015H4014)
文摘An effective design method of freeform micro lens array is presented for shaping varied laser beams into prescribed rectangular illumination. The variable separation mapping is applied to design concave freeform surfaces for constructing a freeform lens array. Several dedicated examples show that the designed freeform optical lens array can achieve a prescribed rectangular illumination pattern, especially without considering the initial states of incident laser beams. Both high collection efficiency and good spatial uniformity can be available simultaneously. Tolerance analysis is also performed to demonstrate that this optical device can well avoid fabricating difficulty in actual applications.
