3.1. Measured Results
The measured Mueller matrices for the laparoscopes are shown in . Each sub-image shows one element of the Mueller matrix across the circular image field of the endoscope. To correct for the illumination intensity the elements are normalized to element m11. Instead of showing the resulting unity image for m11, we have normalized to its own maximum value in the figure to illustrate the illumination field. This shows a radial fall-off, as usually observed in endoscopes due to the uneven illumination and image apodization.
Fig. 3. Measured Mueller matrices for the Olympus and Storz laparoscope (a)–(c) and a simulation for a sheet of sapphire (d). Each sub-image shows one element of the matrix across the whole field. Matrices were measured with the system illustrated in (more ...)
Uniform polarization effects across the field of view would result in elements of uniform value and hence flat images displaying no structure. This is clearly not the case as highly structured patterns are visible across the field of view, meaning the polarization properties change with image co-ordinate. The Olympus endoscope showed circular arcs in elements m22 to m43, which were observed to rotate and change value as the endoscope was rotated about its long axis [compare and ]. The Karl Storz endoscope showed patterns with high amounts of circular symmetry which did not change if it is rotated (hence only one orientation is shown). Both patterns exhibited a simple wavelength dependence, expanding outward as the wavelength increased (Not shown).
Both laparoscopes shared the property that elements m12 to m14 (top-row) and m21 to m41 (left-column) were zero. From this we can deduce that they do not contain polarizing elements and only interact with light that is already polarized. Therefore these effects are not apparent when illuminated using standard sources such as halogen lamps.
3.2. Simulated Results
In order to better understand where these effects may originate we considered the arrangement of rays passing through the entrance and exit windows of the laparoscopes as shown in . Rays originating from an image point are aligned approximately parallel to each other when they pass through the exit window with points at the edge of the image passing at greater angles. This is equivalent to the conoscopic geometry used in crystallography [17
] and illustrated in . The important feature of this arrangement is that it directly interchanges image coordinates for angles passing through a sample. This allows the birefringence of a sample to be easily assessed at all angles, and any symmetry properties of the resulting conoscopic images are related to the orientation and symmetry properties of the crystal lattice. In our case the exit window is in an equivalent plane to the crystal sample.
Fig. 4. The standard conoscopic geometry. Points in the image correspond to angles through the sample, usually a crystal. Any symmetry in the captured images will correspond to symmetries in the crystal lattice of the sample. These will change with the orientation (more ...)
The windows of the Karl Storz laparoscope are made from sapphire which is a birefringent crystal. To confirm that the observed patterns could be explained by this we simulated the Mueller matrix that would be expected in a conoscopic geometry if a thin sheet of sapphire had its slow axis aligned parallel to the optic axis. The half-angle of the field of view for the laparoscope was calculated as approximately 38.6°. A grid of angles matching this was created and used to find the effective refractive indices for the ordinary and extra-ordinary rays, which were subtracted to find the effective birefringence at a wavelength of 600 nm. The resulting Jones matrix was calculated using an assumed thickness of 0.5 mm for the sapphire sheet at all incidence angles. These were pre- and post-multiplied by the Jones vectors corresponding to the measurement polarizations to produce simulated images of a conoscopy experiment. The same technique as for the real experiment was then used to convert these into a Mueller matrix. The Jones calculus was used as it simplifies calculations where only fully polarized light is used. The simulation is presented in which shows closely matching patterns for the top left 3×3 sub-matrix with , however they do not match in the right-most column and bottom row (the elements of a Mueller matrix relating to circular polarizations).
This indicated that the λ/4 waveplates might cause the discrepancy between measurement and simulation as they are only used to measure these elements. Achromatic waveplates have an angular dependence on their retardance and so are not suited to imaging applications. The angular dependence means that at points away from the centre we are no longer dealing with purely circular polarization but some elliptical state. The Karl Storz matrix was decomposed using Lu and Chipman’s technique, shown in . The diattenuation and depolarization were approximately zero across the field of view, as expected. The retardance exhibited angular symmetry with a sinusoidal radial profile. However all of the parameters had X-shaped patterns where they were discontinuous. This effect is strongest at the distal end of the laparoscope as the rays from the object crossing the waveplate are at steeper angles than those at the proximal end.
Fig. 5. The decomposed parameters of the Storz laparoscope (a)–(c) and the calculated retardance of a magnesium fluoride plate (d) for similar viewing angles, showing the same X-shaped patterns in both. Adapted from .
We then simulated the retardance through a sheet of magnesium fluoride, a material used to make achromatic waveplates using the same technique as for sapphire above. The result is shown in and demonstrates the same X-shaped zones. This implies that the waveplates do affect the measurement of the Mueller matrix and hence the mismatch with simulation. Using zero-order waveplates would reduce the angular dependence although they would introduce a wavelength dependence, meaning the results would only be correct close to the central wavelength of the plate.
The close match with the simulation of the sapphire sheet shows that the polarization effects of the Karl Storz laparoscope come primarily from the birefrigent window material. We are currently unable to provide a similarly convincing explanation of the behaviour of the Olympus laparoscope. We are unaware of the exact window material used but the lack of any symmetry means that the windows cannot be made from a birefrigent crystal oriented as a λ/4 or λ/2 waveplate, although the orientation dependence and interaction with circularly polarized light does display waveplate-like properties.
3.3. Possible Solutions
In order to use commercial endoscopes with polarized imaging any polarization effects must be calibrated and corrected. In theory this can be done simply by measuring the Mueller matrix of a particular laparoscope before use in theatre, and provided this matrix is not degenerate its inverse could be post-multiplied by any Stokes’ measurements made through the laparoscope to extract the polarization state at the distal tip. When no circular polarizations are present in a system it is possible to use just a three-by-three Mueller matrix and consider only the linear states. However the birefringent nature of the crystals used means that most rays exiting the endoscope will have a circular component, and this cannot just be ignored. Hence the full Mueller matrix would have to be measured for every pixel in the image and at all wavelengths used in the system. But as described above achromatic waveplates have a strong angular dependency, whereas zero-order waveplates have a strong wavelength dependency. This means that the circular components of the Mueller matrix cannot be measured accurately for all pixels and wavelengths.
A simpler way of removing the polariztion effects would be to remove the birefrigent crystals and replace them with non-birefrigent alternatives. Unfortunately simple fused silica is unsuitable since the bonding agents used with it cannot withstand the autoclave sterilization process. A good alternative would be diamond, as this uses the same bonding materials as sapphire but is not birefringent.