Oblique projection optical system

An oblique projection optical system including at least one decentered lens element, said oblique projection optical system comprising:

• a lens unit having a negative focal length and comprising at least one lens element;
• and
• a lens unit having a positive focal length and comprising at least one lens element,
• wherein an overall focal length of the oblique projection optical system is varied by varying at least one of relative positions of the lens unit having the negative focal length and the lens unit having the positive focal length,
• where the focal length of the oblique projection optical system is defined as follows:
• by using as a reference a ray (base ray) traveling from an object center by way of a diaphragm center toward an image plane, a value ascribed to a given surface (an i-th surface) is defined by a procedure (a), and a composite focal length from a given surface (a k-th surface) to a given surface (an n-th surface) is defined by a procedure (b);
• here, a coordinate system of the i-th surface is a rectangular coordinate system (ξi, η, ζi) defined as follows:
• a direction of a normal to the i-th surface is a ζi axis, a direction vertical to a surface formed by the base ray incident on the i-th surface and the normal to the i-th surface (that is, a plane of incidence) is a ξi axis, and a direction where Qi×Ei is positive is a positive direction;
• Qi is a vector in a direction in which the base ray is incident on the i-th surface, Ei is a vector in the direction of the normal to the i-th surface, and the ηi axis is decided by a right hand system with respect to the ξi axis and the ζi axis;
• 1) procedure (a)
• (ξi direction)
• Fξi=−(di/ni)−{1/(Di·Sξi)},
• Mξi=1/(Di·Sξi), and
• Bξi=−1/(Di·Sξi);
• (ηι direction)
• Fηi=−(di/ni)−{1/(Di·Sηi)}·cos2(θi),
• Mηi={1/(Di·ηi)}·cos(θi)·cos(θ′i), and
• Bηi=−{1/(Di·Sηi)}·cos2(θ′i);
• here
• Di={ni·cos(θi)−n′i·cos(θ′i)},
• Sξi=∂2gi/∂ξ2, and
• Sηi=∂2gi/∂η2,
• where
• di is an actual distance, along the base ray, from a point of exit of an (i−1)-th surface to a point of incidence on the i-th surface,
• ni is a refractive index of a medium of the i-th surface on a side immediately before incidence of the base ray,
• ni′ is a refractive index of the medium of the i-th surface on a side immediately after the incidence of the base ray,
• θi is an angle of incidence of the base ray on the i-th surface,
• θ′i is an angle of exit of the base ray from the i-th surface,
• gi is a surface configuration expression, g=g(ξ, η), of the i-th surface, and
• Sξi and Sηi are values of local curvatures expressed as quadratic differential of g in the ξ and η directions at the point of incidence of the base ray;
• 2) procedure (b)
• the k-th surface and a (k+1)-th surface are synthesized based on the following synthesis expressions:
• M(k)+(k+1)=−Mk{Bk+F(k+1)}−1·M(k+1), and
• B(k)+(k+1)=B(k+1)−M(k+1){Bk+F(k+1))}−1·M(k+1);
• by performing calculation while successively rewriting the synthesis expressions by changing k to (k)+(k+1) and changing k+1to k+2, the k-th to a (k+2)-th surfaces are synthesized and by repeating this calculation to the n-th surface, the composite focal length f of the k-th to the n-th surfaces is defined as follows:
• (ξi direction)
• fξ(k)−(n)=−Mξ(k)−(n), and
• (ηi direction)
• fη(k)−(n)=−Mη(k)−(n);
• in a case of a one-plane-symmetrical optical system in which a Y-Z plane in a rectangular coordinate system (X, Y, Z) is symmetrical, the composite focal length f is defined as follows:
• (X direction)
• fX(k)−(n)=fξ(k)−(n), and
• (Y direction)
• fY(k)−(n)=fη(k)−(n);
• in the following case, the following synthesis expressions are used instead of the above synthesis expressions:
• when Sξk=0 (when a local curvature in the ξ direction is 0, that is, when a local radius of curvature is ∞),
• Mξ(k)+(k+1)=Mξ(k+1), and
• Bξ(k)+(k+1)=Bξk+1);
• when Sηk=0 (when a local curvature in the η direction is 0, that is, when a local radius of curvature is ∞),
• Mη(k)+(k+1)=Mη(k+1){cos(θk)/cos(θ′k)}, and
• Bη(k)+(k+1)=Bη(k+1);
• when Sξ(k+1)=0 (when the local curvature in the ξ direction is 0, that is, the local radius of curvature is ∞),
• Mξ(k)+(k+1)=Mξk, and
• Bξ(k)+(k+1)={Bξk·n(k+1)−d(k+1)}/n(k+1);
• when Sη(k+1)=0 (when the local curvature in the η direction is 0, that is, the local radius of curvature is ∞),
• Mη(k)+(k+1)=Mηk{cos(θ′(k+1))/cos(θ(k+1))}, and
• Bη(k)+(k+1)=[{Bηk·n(k+1)−d(k+1)}·cos2(θ′(k+1))]/{n(k+1)·cos2(θ(k+1))};
• when Sξk=0 and Sξ(k+1)=0 (that is, when a first surface is a parallel plate),
• Mξ(k)+(k+1)=−∞, and
• Bξ(k)+(k+1)=∞;
• considering synthesis of the first to the (k+2)-th surfaces,
• Mξ(k)+(k+1)+(k+2)=Mξ(k+2), and
• Bξ(k)+(k+1)+(k+2)=Bξ(k+2);
• when Sηk=0 and Sη(k+1)=0 (that is, when the first surface is a parallel plate),
• Mη(k)+(k+1)=−∞, and
• Bη(k)+(k+1)=∞;
• considering synthesis of the first to the (k+2)-th surfaces,
• Mη(k)+(k+1)+(k+2)=Mη(k+2), and
• Bη(k)+(k+1)+(k+2)=Bη(k+2).