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High resolution extended image near field optics:

2. An idealised symmetric extended image near field imaging device

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Copyright (c) Malcolm Kemp 2010

 

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Consider a rotationally symmetric optical layout with an (axial) cross-section as per Figure 1. This consists of two large highly elongated truncated ellipsoidal mirrors, with plane mirrors (perpendicular to the axis of rotation) placed at the left and right hand ends of the arrangement. The centre of the mirror at the left hand end of the layout, , is one of the focal points of the ellipsoid that forms the left half of the layout. The centre of the mirror at the right hand end of the arrangement, , is one of the focal points of the ellipsoidal mirror that forms the right half of the layout. Both ellipsoidal mirrors also share a focal point at , half-way along the layout (i.e. the two ellipsoidal mirrors are confocal).  forms a straight line, so the ellipsoidal mirrors are also coaxial.

 

 

Figure 1: A large highly elongated symmetric truncated confocal and coaxial ellipsoidal mirror pair, with plane mirrors perpendicular to the axis of rotation at each end of the layout

 

Suppose that:

 

(a)    The plane mirrors at each end of the layout are thin ‘nearly fully silvered’ idealised reflectors (with the silvering pointing inwards, i.e. in each case towards ), i.e. they transmit a small fraction of light incident on them but otherwise perfectly reflect all of the light incident onto them);

 

(b)   Both ellipsoidal mirrors are ‘partly silvered’ idealised reflectors, i.e. perfectly reflect a proportion of all of the light incident onto them. It is assumed that behind them is a perfect absorber, so that any light transmitted through them can be ignored. The extent to which they need to be partially rather than fully silvered depends on the extent to which light that has bounced back and forth between the plane mirrors would corrupt the image formation. Image blurring arising because of these path trajectories can be eliminated by making the ellipsoidal mirrors only slightly mirrored, but at the expense of less light being available to create the image;

 

(c)    All four mirrors are many wavelengths in size;

 

(d)   The ellipsoidal mirrors are arbitrarily elongated (so the angle subtended by the hole at  on either  or  is arbitrarily small);

 

(e)   A flat object is placed a small fraction of a wavelength to the left of ; and

 

(f)     The object , whilst many wavelengths in size, is only an arbitrarily small fraction of the size of the entire aperture formed by the rim of the truncated ellipsoidal mirror (so is not drawn to scale in Figure 1).

 

What image of the object in (e) would be formed a small fraction of a wavelength to the right of ?

 

In the absence of the two end plane mirrors, the ellipsoidal mirror pair form an aplanatic layout, with object and image planes at  and  respectively. We would therefore expect it to create a clean, but Rayleigh resolution-limited, image at  of the object placed at  in a manner similar to any other ‘conventional’ imaging arrangement. For the image not to suffer material amounts of spherical aberration, we need the object to be small relative to the distance between the focal point and the nearest rim of the ellipsoidal mirror, but given design feature (e) the object could still be many wavelengths in size before this became an issue. Objects placed a sufficiently small fraction of a wavelength behind at  would therefore form an image a sufficiently small fraction of a wavelength behind  that is arbitrarily close in form to a conventional Rayleigh resolution-limited image.

 

However, there are three ways in which the complete layout described in this hypothetical situation differs from that a ‘conventional’ imaging arrangement:

 

(i)      The nearly fully silvered plane mirror at the right hand end of the layout converts the device from a far-field to a near-field device. There is now an active part of the device near to, indeed exactly in the image plane;

 

(ii)    The layout subtends a solid angle onto the image plane at  that is almost the maximum possible onto a plane. The only rays that are missing from the complete span of possible ray trajectories are ones that would otherwise have been coming from the vicinity of . Design feature (d) means that these form an arbitrarily small proportion of the total angle span onto the image plane and so in the limit can be ignored; and

 

(iii)   The nearly fully silvered plane mirror at the left hand end of the layout constrains the nature of the light waves entering the cavity formed by the ellipsoidal mirrors, and thus also constrains the nature of the waves converging onto the image plane.

 

Our assertion is that inclusion of these non-standard aspects to the layout result in an image of  being formed at  that is no longer subject to the Rayleigh resolution limit. Indeed the image should be arbitrarily accurate, to the extent that it is possible to create such an idealised layout in practice. Moreover, if the plane mirrors are sufficiently close to being fully silvered as per design feature (a), then the device would create an extended image that circumvents the Rayleigh resolution limits that might make SNOM-type technology more commercially viable.

 

Readers may, however, object that, even if this assertion were true, the design features needed for the above device to work would involve some carefully crafted limiting properties. Some of these relate to the physical characteristics of the materials used to make the mirrors, some relate to the dimensions of the layout (both width and length) relative to the object being imaged and some relate to the proportion of light leaving the object that is reconstituted to form the image. Moreover, with the above design the object and image are of the same size, limiting the practical usefulness of the proposed design for microscopy and certainly rendering it useless for telescopy.

 

The main aim of discussing the above layout is thus to elucidate the principles involved and to suggest ways in which the layout would need to be refined were it to be applied in practice.

 


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