| 1. A resonator, comprising: a three-dimensional photonic crystal; and a period defect member provided in said three-dimensional
photonic crystal, wherein said three-dimensional photonic crystal includes a periodic structure member having periodically
laminated layers including a refractive index periodic structure, wherein said periodic structural member includes (i) a first
layer having a periodic structure being provided with holes which comprise a second medium and which are formed at lattice
points of a first rectangular lattice having a period a along a first axis extending along a plane of the first layer and
a period b along a second axis extending along the plane of the first layer and being perpendicular to the first axis, and
at lattice points of a second rectangular lattice defined at a position determined by shifting the first rectangular lattice
by a/2 along the first axis and by b/4 along the second axis, wherein a region of the first layer other than the holes is
filled with a first medium, (ii) a second layer having a periodic structure with a columnar structure which comprises a third
medium and has an axis extending in a layer lamination direction and which is formed at lattice points of a face-centered
rectangular lattice defined at a position determined by shifting the first rectangular lattice by +3b/8 along the second direction
and having a period a along the first axis and a period b along the second axis, wherein a region of the second layer other
than the columnar structure is filled with the second medium, (iii) a third layer having a periodic structure formed by providing
the periodic structure included in the first layer at a position shifted by a/2 along the first axis and by b/2 along the
second axis relative to the first layer and with respect to a direction along the plane of the second layer, and (iv) a fourth
layer having a periodic structure formed by providing the periodic structure included in the second layer at the same position
as the second layer with respect to a direction along the plane of the second layer, wherein the first, second, third and
fourth layers are laminated in this order, and wherein said periodic defect member includes a columnar structure provided
at the second layer or the fourth layer and on an axis of the columnar structure formed in the second layer or the fourth
layer; wherein the holes formed in the first and third layers of said three-dimensional photonic crystal have a circular sectional
shape along the plane of the layers, and wherein said resonator is operable in a single mode.
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