We report the capture of images via a wedge light-guide without the margin for fan-in needed heretofore. While this lets one look out of a slim panel as if it were a periscope, half the power is lost and resolution is degraded by aperture diffraction. Volume gratings may resolve these drawbacks at certain wavelengths and we consider how these might be extruded.
© 2010 OSA
One typically expects to open a book in order to read it or that space is needed either side of a lens if it is to form an image. However, it was recently shown that an image can be captured or projected through a slim wedge-shaped light-guide [1
1. A. R. L. Travis, F. P. Payne, J. J. Zhong, and J. R. Moore, “Flat panel display using projection within a wedge-shaped waveguide,” presented at the 20th International Display Research Conference, Society for Information Display, 292–295 (2000).
2. Y. K. Cheng, S. N. Chung, and J. L. Chern, “Aberration analysis of a wedge-plate display system,” J. Opt. Soc. Am. A 24(8), 2357–2362 (2007). [CrossRef]
]. Such a light guide has many of the properties of a lens where the thick end constitutes an almost one-dimensional focal plane. Nevertheless, between the thick end and the wedge surface is needed a slab light-guide within which rays fan into (or out from) the thick end. This slab constitutes a margin which is undesirable in such a system where the aim is to be compact.
If it is enough merely to collimate illumination from a point source, the margin can be eliminated by injecting the rays at the thin end of a blunt wedge so that they fan out towards the thick end off which they reflect [3
]. The thick end is embossed with mirror facets which increase ray angle relative to the plane of the wedge so that as they return towards the thin end, the rays emerge. But wedge light-guides are also used to collect the optical image from the phosphor plate of an X-ray scanner [4
4. S. Boual, T. Large, M. Buckingham, A. R. L. Travis, and S. Munford, ““Wedge displays as cameras”, in SID International Symposium,” Dig. Tech. Pap. 37, 1999–2002 (2006). [CrossRef]
] and margin is also a nuisance in this application. Here we explain why the angle of the facets must differ from that for collimation if the system is instead to be used for the projection or capture of images. We also explain why a different method of reflection will be required if we are to avoid the loss of power and resolution which facets introduce.
Consider a ray of light shone into the thick end of a transparent wedge as shown in Fig. 1
Fig. 1 The slab lets rays fan out in the x-z plane. The slab also makes all rays undergo the same number of reflections before exit, so conversion of angle to distance is continuous
. At each reflection within the wedge, the y
angle of the ray relative to the normal of the opposite surface will diminish until the critical angle is reached at which point the ray will emerge into air. The number of reflections required to reach this point will depend on the angle of injection so this angle is converted by the wedge into distance along the z
-axis in the same manner as with projection through free space.
The projected image should also expand to fill the width of the screen along the x
-axis so there is a slab between the point of input and the wedge in Fig. 1
. Rays fan-out within the x
plane of the slab but the slab does more than just allow fan out. Its length along the z
-axis is carefully chosen to meet the design rule that all rays undergo the same number of reflections before exit, for example each ray in Fig. 1
undergoes seven reflections. This is so that as the input angle of a ray is increased, there is a smooth increase in the distance to the ray’s point of exit and there is no sudden jump which an extra reflection would produce. Unfortunately a kink appears on the top (maximum y
co-ordinate) surface of Fig. 1
where the slab meets the wedge and while the kink can be smoothed into a curve, the curve is cylindrical which means that it introduces astigmatism to the projected image.
As is the case for a collimating wedge [3
], the slab of an imaging wedge can be eliminated by injecting light at the thin end of a blunt wedge. Rays fan out to the thick end which is spherically curved so that they reflect in parallel as shown in Fig. 2
Fig. 2 No margin is needed if rays can fan out within the wedge. We do this by pointing the camera or projector into the thin end.
. The thick end is made twice as thick as the thin end so that, according to the Lagrange invariant, the angle of incident rays relative to the plane of the wedge at the thick end is half the maximum. Furthermore, the thick end should be configured to deflect all incident rays to angles greater than half the maximum whereupon rays are obliged to emerge as they travel back towards the thin end.
All rays are approximately at the critical angle as they leave the wedge so they can be traced backwards in parallel from the final surface. We trace them through a stack of reflections of the wedge such as presents itself to a ray within the light-guide in the manner of a kaleidoscope. The thick end of the wedge is curved slightly so that the end of the stack has a constant radius of spherical curvature. This focuses parallel rays towards a point halfway towards the centre of curvature, i.e. to a point at the thin end. The number of reflections undergone by any ray is equal to the number of surfaces crossed by the trace of the ray in the stack. By virtue of focus therefore, all rays undergo the same number of reflections in the manner required by our design rule for the projection of an unbroken image.
It might seem that the simplest way to increase the angle of rays incident from the point of focus as they reflect off the thick end is to emboss it with the saw-tooth of facets shown in Fig. 3
Fig. 3 The wedge thick end is curved so when traced backwards, rays form a focus whose position can be altered by facets.
. But rays are as likely to hit the thick end after hitting the top surface of the wedge as the bottom so the end structure must be symmetric. We therefore embossed the thick end with a zig-zag of facets, the same as used for collimation [3
] but with a different facet angle as calculated in section 3. The penalty for this choice is that half the rays are lost to the system and the remainder subject to aperture diffraction.
A polymethyl methacrylate wedge was prepared similar to that used for collimation [3
] i.e. a linear taper from 6.2 mm to 10.8 mm over a distance of 320 mm with a spherical curve machined on the thick end. Against the thick end we glued a zig-zag mirror extruded perpendicular to the diagram of Fig. 3
. The angle of the lines forming the zig-zag relative to the local plane of the curve was one quarter of the difference between 90° and the critical angle. Prismatic turning film was placed adjacent to one of the wedge surfaces such that as rays from the central point of focus departed the wedge, they were turned perpendicular to its surface. A camera was pointed into the centre of the thin end and a back illuminated test-card was placed against the free surface of the wedge and the result is shown in Fig. 4
Fig. 4 Image captured by a camera pointed into the thin end of a wedge with a test card placed against one surface.
4. Properties of the reflector
The zig-zag of facets degrades resolution and wastes light so is there another way we can configure our system? Returning to the stack of wedges, it is important to note that to an observer within the wedge, the surfaces are only apparent when the critical angle is reached and the stack otherwise behaves like a solid block of material. The angle of the facets at the thick end can be altered in synchrony so as to swivel the point of focus up and down the left hand side of the diagram. Two points interest us: the point of focus shown in Fig. 5
A second arrangement permits imaging where the angle of the facets is twice that in Fig. 3
which is at the opposite extreme to that of Fig. 3
, and the point of focus half-way between these extremes which was that used for the collimating wedge [3
puts the focus at the opposite extreme to that of Fig. 3
and we see that the ray injected near the critical angle emerges near the thin end and therefore undergoes many more reflections than the ray emerging near the thick end. It is nevertheless clear from the analogy with bulk material that there will be no discontinuities in the projected image - perhaps our design rule should be that the number of bounces off the top surface less that off the bottom should be constant for all rays. Note however that the orientation of the projected image will be opposite to that of Fig. 3
for which the ray injected near the critical angle emerges near the thick end. It follows that none of the focal points in between the extremes of Fig. 3
and Fig. 5
is useful for the projection of images: there would be a superposition of projected images of opposite sense. The significance of the half-way point is that this can be used to make a collimated backlight without power loss but it is no good for image projection or capture.
Turning now to the facets, aperture diffraction and loss of resolution might be eliminated, at least for light of one wavelength, if the reflector were more like a Bragg grating. It is well known that layers of alternating refractive index spaced at half a wavelength will reflect normally incident light and that the angle of reflection can be altered by varying the layer spacing. Superpose in the manner of Fourier theory the spatial periods appropriate to a range of angles and the index versus depth that results is shown in Fig. 7
Fig. 7 A multi-layer grating was modeled in which the layers were tilted at 15° and varied in refractive index in the manner of a rugate filter.
. This is a rugate filter [5
5. H. A. Macleod, Thin-Film Optical Filters, (Institute of Physics Publishing, 2002), Chap. 14.1.
] which reflects over a range of angles while being transmissive outside these angles. If we could get the same properties when the layers were tilted, then we might tilt the layers by e.g. 15°, design them to reflect at ± 15° to the layer perpendicular and thereby reflect light incident at 0° to 30° while transmitting light incident at 0° to −30° as illustrated by Fig. 6
Fig. 6 Target properties of the end reflector.
. The idea is that one might then place a second filter behind with layers tilted at −15° so as appropriately to reflect the rays transmitted by the first.
The reflection off sloped dielectric layers with a constant amplitude of index modulation can be found analytically [6
6. R. S. Chu and J. A. Kong, “Modal Theory of Spatially Periodic Media,” IEEE Trans. Microw. Theory Tech. MTT-25, 18–24 (1977).
] but a rugate filter is less simple. By way of a preliminary study, finite element analysis was used to model the reflection of light of wavelength 532 nm off a 43 μm deep stack of dielectric layers sloped at 15° to the plane of the stack and with a variation in refractive index of as much as 0.1 about an average index of 1.5195 as shown in Fig. 7
. The reflection versus angle of incidence is shown in Fig. 8
The properties predicted by finite element analysis of the multi-layer grating of Fig. 7
[values greater than 1are presumed due to software artifacts].
where the anomalous reflectance of >l is thought to be an artifact of the software with reflectance more likely close to unity at the negative angles of incidence shown.
A pair of such stacks was superposed, one sloped in opposite sense to the other, and the system as a whole was subject to finite element analysis. The stacks were found to act without interference of one another.
The advantage of injection at the thin end is not only that no margin is needed for fan-out. The radius of curvature at the thick end is spherical so this system is not subject to the astigmatism of the design of Fig. 1
. Furthermore, the image can go right to the edge of the thick end and if the projector is folded underneath the thin end, there is the potential for a display or camera system where the picture totally fills the screen. Lastly, aside from the facets, the active surfaces of the device are two flats and a sphere which are easily ground from glass which can cost less and have less scatter than plastic alternatives.
The principle of laminating and extruding many hundreds of layers into a tape [7
7. M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287(5462), 2451–2456 (2000). [CrossRef] [PubMed]
] or fiber [8
] is well-established but the index range used in our simulation is wider than the 1.45 to 1.58 range demonstrated in graded index plastics [9
9. Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New Class of Bioinspired Lenses with a Gradient Refractive Index,” J. Appl. Polym. Sci. 103(3), 1834–1841 (2007). [CrossRef]
]. Even with a full range of indices, we are far short of the tantalizing ideal of an end reflector as defined by Fig. 6
which reflects white light so that this device could be used without caveat.
We have demonstrated the capture via a slim light-guide of images placed in front of any part of that light-guide. The equivalent device working in the reverse direction with a video projector would give a flat panel display with no margins. Half the power is lost and aperture diffraction is introduced by the facet reflectors presently used at the thick end of the light-guide. Finite element analysis has predicted that a multi-layer reflector such as might in principle be made by extrusion can eliminate the loss and aperture diffraction for light of one wavelength.
References and links
A. R. L. Travis, F. P. Payne, J. J. Zhong, and J. R. Moore, “Flat panel display using projection within a wedge-shaped waveguide,” presented at the 20th International Display Research Conference, Society for Information Display, 292–295 (2000).
Y. K. Cheng, S. N. Chung, and J. L. Chern, “Aberration analysis of a wedge-plate display system,” J. Opt. Soc. Am. A 24(8), 2357–2362 (2007). [CrossRef]
A. R. L. Travis, T. Large, N. Emerton, and S. Bathiche, “Collimated light from a waveguide for a display backlight,” Opt. Express 17(22), 19714–19719 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=opex-17-22-19714. [CrossRef] [PubMed]
S. Boual, T. Large, M. Buckingham, A. R. L. Travis, and S. Munford, ““Wedge displays as cameras”, in SID International Symposium,” Dig. Tech. Pap. 37, 1999–2002 (2006). [CrossRef]
H. A. Macleod, Thin-Film Optical Filters, (Institute of Physics Publishing, 2002), Chap. 14.1.
R. S. Chu and J. A. Kong, “Modal Theory of Spatially Periodic Media,” IEEE Trans. Microw. Theory Tech. MTT-25, 18–24 (1977).
M. F. Weber, C. A. Stover, L. R. Gilbert, T. J. Nevitt, and A. J. Ouderkirk, “Giant birefringent optics in multilayer polymer mirrors,” Science 287(5462), 2451–2456 (2000). [CrossRef] [PubMed]
M. Mignanelli, K. Wani, J. Ballato, S. Foulger, and P. Brown, “Polymer microstructured fibers by one-step extrusion,” Opt. Express 15(10), 6183–6189 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=opex-15-10-6183. [CrossRef] [PubMed]
Y. Jin, H. Tai, A. Hiltner, E. Baer, and J. S. Shirk, “New Class of Bioinspired Lenses with a Gradient Refractive Index,” J. Appl. Polym. Sci. 103(3), 1834–1841 (2007). [CrossRef]
(050.7330) Diffraction and gratings : Volume gratings
(080.3630) Geometric optics : Lenses
(110.2990) Imaging systems : Image formation theory
(230.7390) Optical devices : Waveguides, planar
Original Manuscript: January 22, 2010
Revised Manuscript: April 4, 2010
Manuscript Accepted: April 5, 2010
Published: April 7, 2010
Adrian R. L. Travis, Tim Large, Neil Emerton, Zhaoming Zhu, and Steven Bathiche, "Image capture via a wedge light-guide with no margins," Opt. Express 18, 8453-8458 (2010)