Ultra-high resolution imaging devices

A method of designing a two-mirror high numerical aperture imaging device comprising the steps of:

• (a) determining the position of each consecutive point of a cross-section through the x-axis of a first mirror located nearest an object for imaging in use and a second mirror located nearest the image in use, iteratively for a cross-section in the plane z=0, where the x and y coordinates of each successive point on the first and second mirrors in the cross-section are determined by the functions M1(t)≡(mi,x(t), m1,y (t),0) and M2(t)≡(m2,x(t), m2,y(t),0) respectively, t being an iteration counter;
• where the functions M0(t)≡(m0,x(t), m0,y(t),0)≡(f,0,0) and M3(t)≡(m3,x(t), m3,y(t),0)≡(0,0,0) are set to define the centres of an object plane and an image plane, respectively;
• where a1(t) and a2(t) are the angles that tangents to the first and second mirrors make to the z-axis, and a0(t) and a3(t) are the angles that the object and image planes make to the x-axis (being 90° for all t for the mirrors to be rotationally symmetric about the x-axis);
• di(t) (for i=0, 1, 2) is the angle that a ray from Mi(t) to Mi+1(t) makes to the x-axis and pi(t) (for i=0, 1, 2) is the distance between Mi(t) and Mi+1(t), so that:
• d i ⁢ ⁢ ( t ) = arctan ⁢ ⁢ ( m i + 1 , y ⁢ ⁢ ( t ) - m i , y ⁢ ⁢ ( t ) m i + 1 , x ⁢ ⁢ ( t ) - m i , x ⁢ ⁢ ( t ) ) a i ⁢ ⁢ ( t ) = d i ⁢ ⁢ ( t ) 2 + d i - 1 ⁢ ⁢ ( t ) 2 p i ⁢ ⁢ ( t ) = ( m i + 1 , x ⁢ ⁢ ( t ) - m i , x ⁢ ⁢ ( t ) ) 2 + ( m i + 1 , y ⁢ ⁢ ( t ) - m i , y ⁢ ⁢ ( t ) ) 2
• (b) selecting suitable parameters to define the size and shape of the device—for use with a far away source (of the order of f=−109 distant along the negative x-axis) define the overall width of the device by the value of B, B=b/p0(0);
• define two further parameters m2,y(0)=k and m2,x(0)=−q, and taking m1,x(0)=m2,x(0)−δ, where δ is small or 0 (thereby defining one limit of the acceptable range of iterated values);
• (c) assigning values to m1,y(0) and Z:
• Z = 1 ⁢ ⁢ m 1 , y ⁢ ⁢ ( 0 ) = - m 2 , y ⁢ ⁢ ( 0 ) ( m 2 , x ⁢ ⁢ ( 0 ) 2 + m 2 , y ⁢ ⁢ ( 0 ) 2 ) 1 / 2
• (d) updating the values of M1(t) and M2(t) according to the following iterative formulae for a small value of h, where wi(t) is a function corresponding to the distance between Mi(t) and Mi(t+1) and ri−1(t)=sin(a−1(t)−di−1(t)) si(t)=sin (ai(t)−di−1(t)) M i ⁢ ⁢ ( t + 1 ) ≡ ( m i , x ⁢ ⁢ ( t + 1 ) m i , y ⁢ ⁢ ( t + 1 ) ) = M i ⁢ ⁢ ( t ) + w i ⁢ ⁢ ( cos ⁢ ⁢ ( a i ⁢ ⁢ ( t ) ) sin ⁢ ⁢ ( a i ⁢ ⁢ ( t ) ) ) ⁢ ⁢ h
• where the distance parameters wi(t) for w1, w2, a1′w1 and a2′w2 satisfy the four simultaneous equations:
• w2h=E1(w1h,0)0=E2(w2h,w1h) w2h=E1(0,h) ZBh=E2(w2h,0)
• (h corresponding to the distance away from the centre that rays eventually strike the image plane a distance ZBh from its centre and ai′(t)gi is the difference between the angle that the i′th mirror makes to the x-axis at point Mi(t) and the angle that it makes at a distance gi from Mi(t), for a smooth surface this being a linear function of gifor small gi, ai′(t) being the derivative of ai(t) with respect to t);
• w1(t) and w2(t) further satisfying the relationships:
• w 2 ⁢ ⁢ h = E 1 ⁢ ⁢ ( 0 , h ) ⁢ ⁢ p 1 ⁢ ⁢ r 0 ⁢ ⁢ h p 0 ⁢ ⁢ s 2 ⁢ ⁢ 0 - ZBh = ( E 2 ⁢ ⁢ ( w 2 ⁢ ⁢ h , w 1 ⁢ ⁢ h ) - E 2 ⁢ ⁢ ( w 2 ⁢ ⁢ h , 0 ) ) = p 2 ⁢ ⁢ r 1 p 1 ⁢ ⁢ s 3 ⁢ ⁢ w 1 ⁢ ⁢ h
• where Ei(gi, gi−1) represents the distance from Mi+1(t) that a ray will strike the i+1′th surface if it comes from a point on the i−1′th surface that is a distance gi−1 from Mi−1(t) via a point on the i′th surface that is a distance gi from Mi(t), for small gi−1 and gi, such that for deflection by reflection:
• E i ⁢ ⁢ ( g i , g i - 1 ) = ( r i s i + 1 - p i ⁢ ⁢ s i p i - 1 ⁢ ⁢ s i + 1 ) ⁢ ⁢ g i + ( p i ⁢ ⁢ r i - 1 p i - 1 ⁢ ⁢ s i + 1 ) ⁢ ⁢ g i - 1 + ( 2 ⁢ p i s i + 1 ) ⁢ ⁢ a i ′ ⁢ ⁢ g i i . e . ⁢ w 2 = p 1 ⁢ ⁢ r 0 p 0 ⁢ ⁢ s 2 ⁢ ⁢ w 1 = - ZB ⁢ ⁢ p 1 ⁢ ⁢ s 3 p 2 ⁢ ⁢ r 1
• (e) ending the iteration when m2,x(0) approaches zero thereby defining the other limit of the acceptable range of iterated values;
• (f) rotating the curves thus produced, which define the shapes of the two mirrors, around the x-axis to define a complete, three-dimensional two-mirror arrangement.