Straightforward numerical integration of the Rayleigh-Sommerfeld diffraction integral (R-SDI) remains computationally challenging, even with today’s computational resources. As such, approximating the R-SDI to decrease the computation time while maintaining a good accuracy is still a topic of interest. In this paper, we apply an approximation for the R-SDI that is to be used to propagate the field exiting a Coherent Fiber Bundle (CFB) with ultra-high numerical aperture (0.928) of which we presented the design and modal properties in previous work. Since our CFB has single-mode cores with a diameter (550nm) smaller than the wavelength (850nm) for which the CFB was designed, we approximate the highly divergent fundamental modes of the cores with real Dirac delta functions. We find that with this approximation we can strongly reduce the computation time of the R-SDI while maintaining a good agreement with the results of the full R-SDI. Using this approximation, we first determine the Point Spread Function (PSF) for an ‘ideal’ output field exiting the CFB (identical amplitudes for cores on a perfect hexagonal lattice with the phase of each core determined by the appropriate spherical and tilted plane wave front). Next, we analyze the PSF when amplitude or phase noise is superposed onto this ‘ideal’ field. We find that even in the presence of these types of noise, the effect on the central peak of PSF is limited. From these types of noise, phase noise is found to have the biggest impact on the PSF.
© 2013 Optical Society of America
Until recently, Coherent Fiber Bundles (CFB) were primarily used as biomedical endoscopes. Their small outer diameter and flexibility made them easy to integrate into minimally invasive surgical tools making CFBs ideal for imaging tissue in difficult to reach places which would otherwise require more invasive and painful access if conventional techniques were to be used. With the advent of CCD technology, which offers a better resolution-to-size ratio , CFBs were no longer the prime technology for endoscopic imaging and research concerning the use of CFBs in endoscopy shifted its focus towards non-linear techniques such as OCT [2–5], Raman imaging  and two-photon microscopy . One advantage endoscopic CFBs have over their CCD counterparts is that scanning over the Field-Of-View (FOV) can be achieved by sequentially coupling light into individual cores. This way, each element of the image plane is illuminated sequentially thereby removing the need for additional micro-mechanics  on the distal part of the endoscope. Coupling light into individual cores, however, is challenging [9,10] since individual cores of commercially available CFBs usually have a diameter of 1-3 μm. An alternative optical fiber based scanning technique, investigated by several groups in recent years [11–13], spatially modulates the light at the proximal end (outside the patient) in order to produce a spatially coherent output field at the distal end (inside the patient) thereby mimicking the effect of distal micro-optics. This technique of Proximal Spatial Light Modulation (PSLM) has the advantage that the whole image plane within the FOV can be raster scanned with a focused beam without distal micro-optics and yet without the tissue having to be in direct contact with the optical fiber . In previous work we investigated the requirements for CFBs to be used with PSLM and, based on these requirements, we designed two custom CFBs  (one with focusing and scanning capabilities and one with focusing only) optimized for use with PSLM ([11,12] and  made use of commercially available multi-mode fibers and CFBs respectively). In subsequent research, we focused on the CFB design which would allow for scanning and focusing. This CFB consists of small and closely packed step index, single mode cores (circular) in a common cladding on a hexagonal lattice. The design parameters of this CFB are summarized in Table 1. Note that the ultra-high NA is necessary to ensure adequate light confinement in the cores which are small and closely packed for scanning of the focus over angles of 40° (full cone angle) or more.
We fabricated several prototypes of the CFB according to our design in Table 1 at the Institute of Materials Technology in Warsaw. Using SEM images of the fabricated prototype which best matched our design, we quantified the variations in core size, core shape (or ellipticity) and lattice constant due to the limitations of the fabrication technology and analyzed the influence these variations have on the requirements for the necessary proximal input field in order to achieve a desired field at the output .
The advantage of PSLM is that, within the limits of the fiber’s guided (eigen-)modes, any distal output field can be generated given the correct input field. For example, if the light exiting the CFB needs to be focused at an off-axis point in the image plane, the proximal input field will be spatially modulated in such way that the wave front of the distal output field will be the combination of the appropriate spherical and tilted plane wave front. In order to numerically characterize the Point Spread Function (PSF) of our CFB, different distal output fields (with the same spherical wave front but with different tilted plane wave fronts to simulate scanning of the beam) need to be propagated from the distal exit facet of the CFB towards the image plane. Since the CFB’s single-mode cores have a small diameter-to-wavelength ratio (0.647), their Gaussian fundamental mode will be highly divergent and thus non-paraxial. Accurate propagation of these non-paraxial Gaussians can be achieved via a rigorous technique such as the Rayleigh-Sommerfeld diffraction integral (R-SDI). Since a CFB typically contains thousands of cores, adequate spatial sampling of each Gaussian fundamental mode leads to a large matrix representing the CFB’s distal output field (or object field) to be propagated. Direct integration of the R-SDI, even with a more coarsely sampled image plane, would result in unwieldy computation times. And while FFT based implementations of the R-SDI have a much lower computation time, they do not allow the pixel pitch and field size to be customized independently for the object and image plane . In our case, the small diameter-to-wavelength ratio for the single-mode cores can be used to our advantage as we found that it allowed us to approximate the fundamental mode of each core by a real Dirac Delta function with a certain phase. This allows direct integration of the R-SDI with an object field with Nc points (with Nc the number of cores) instead of N × N points with Nc obviously smaller than N × N. This approximation of the non-paraxial fundamental mode by a Dirac Delta function (and its beneficial consequences for the R-SDI) is further explored in section 2 where we validate it by comparing the PSF predicted by the exact R-SDI with the one predicted by our approximation. In section 3 we use the approximated R-SDI to analyze the influence of different kinds of ‘noise’ on the PSF. In theory, if at the proximal input the correct field is coupled into the CFB, at the CFB’s distal output all the cores will have the same amplitude and the wave front will be the appropriate combination of a spherical and tilted plane. However, when the actual proximal input field differs from this required proximal input field, the actual distal output field will have an amplitude and/or phase different from that of the desired distal output field and this in turn will have consequences for the PSF. In section 3 we compare the PSFs resulting from the propagation of the noise free distal output field for different viewing angles within the FOV with the PSFs resulting from the propagation of ‘noisy’ distal output fields for the same viewing angles and for varying amounts of noise on amplitude, wave front or both.
2. Approximation for the Rayleigh-Sommerfeld propagation formula
The field exiting our CFB consists of the fundamental modes of the cores (on a perfect hexagonal lattice) each with a specific phase determined by the proximal input. In this section, we assume the proximal input field is such that at the output the phases of the cores are determined by the appropriate combination of a spherical and tilted plane wave front so as to allow focusing of the light at any point of our choosing in the image plane within the FOV. The manner in which the appropriate wave front dictates the necessary phase of the cores at the CFB’s distal output, is illustrated in Fig. 1 of the accompanying paper , “part I: modal analysis”. What’s more, in previous work  we have shown that if at the proximal end of the CFB the appropriate field is coupled into the CFB, the field at the distal end will be almost perfectly linearly polarized allowing us to use scalar diffraction theory for the propagation of the CFB’s output field (we refer the reader to  for a more elaborate discussion on the requirements imposed on the proximal input field.). Determining the PSF and FOV of the CFB is now a matter of propagating these fields (assuming the medium surrounding the CFB to be air) towards the image plane using the well-know R-SDI. Unfortunately, for the fields we need to propagate, no closed form analytical solutions for the R-SDI exists. Straightforward numerical integration of the R-SDI, though accurate, has as a major drawback that its computation time scales with NoNi (with No and Ni the number of points in the object and image plane respectively). Adequate spatial sampling of the fundamental mode of each core (say 31 × 31 points per core) in combination with the large number of cores (1951 in our model) in the CFB results in a large No. Implementing the diffraction integral in a straightforward way with such No would then result in unwieldy computation times even with a more coarsely sampled image plane. Therefore, in order to minimize the computation time, we used an approximation based on the fact that the diameter-to-wavelength ratio (0.647) for the cores is smaller than 1 giving them a highly divergent fundamental mode. Usually, the fundamental mode is approximated by a Gaussian function and its free space propagation can subsequently be done using the standard Gaussian beam propagation methods under the paraxial approximation. However, for Gaussian beams with a beam waist smaller than the wavelength, the paraxial approximation requires corrections [20–22]. Corrections, which in effect, amount to considering the single mode cores as complex point sources represented by a Dirac delta function with complex argument of the form23,24]. It is always possible to define, without loss of generality, the coordinate axes in such way that zc = 0 and so that Eq. (1) is reduced to:25]. For such a mode field diameter the Rayleigh length zR is 2μm. Since in endoscopic applications of PSLM the propagation distance z between the plan parallel distal exit facet of the CFB (the object plane) and tissue of interest (the image plane) is at least several hundreds of micrometers, the complex term jzR in Eq. (2) can be neglected. The fundamental mode of each single mode core in the object plane can thus be represented by a Dirac Delta function of the form26]:Eq. (3) in (4) we can calculate the field in the image plane due to the field of a single core, as beingEq. (5)) towards an image plane located at a distance of 500μm. For the conventional R-SDI, the object field consisted of 1281 × 761 sampling points (along x and y direction respectively). We also assumed that the field to be propagated is linearly polarized and consists of the superposition of 19 Gaussians representing the fundamental modes of 19 circular cores (each with diameter 550nm) on a hexagonal lattice with lattice constant Λ = 1500nm. Moreover, each fundamental mode was given a phase ϕr determined by the sampling of a wave front consisting of the sum of a spherical and a tilted plane wave front (see Fig. 1 from ). The spherical wave front was aligned with the optical axis through the center of the object plane in order to focus the propagated field onto the image plane. Also, different plane wave fronts with increasing degrees of tilt (corresponding with the FOV half-angles θ = 0°, 5°, 10°, 15°, 20°, 25°, 30°) were used in order to steer the focus along the x-axis to off axis points in the image plane. The amplitude of each fundamental mode along with the different core phases ϕc for θ = 0°, θ = 15° and θ = 30° are shown in Figs. 1(a)-1(d).
Results of the propagation with both methods are shown in Fig. 2 which shows the cross sections of the propagated fields along the x and y-axis for θ = 0°, θ = 15° and θ = 30°. In general, there is a good agreement between the fields propagated with both methods although there is noticeable decrease in the accuracy of the approximation as θ increases.
One disadvantage of this approximation is that variations in beam divergence, caused by variations in core area (which are present due to the limitation of the fabrication technology as shown in ), cannot be taken into account since the approximation assumes all the cores are point sources. To determine if Eq. (5) would still be a good approximation in the presence of realistic core size variations, we took the fields as shown in Fig. 1 and gave the (circular) cores different diameters according to a Gaussian probability density function with average 550nm and standard deviation 50nm (chosen to be larger than the actual standard deviation of 0.013μm observed in SEM images of fabricated prototypes ). The amplitude of the resulting E-field is shown in Fig. 3. The phase of each core remained the same as described earlier (see Figs. 1(b)-1(d)). Using the standard R-SDI, we again propagated this field towards the observation plane at z = 500μm and compared it with the propagation obtained via the approximation. The resulting cross-sections along the x-axis and y-axis in the image plane for the different values of θ are shown in Fig. 4 where again we see good agreement between the fields propagated with both methods as well as the decrease in the accuracy of the approximation as θ increases.
To quantify the difference between the fields propagated with the exact and approximated R-SDI for both the case with identical cores as well as with variable core sizes, we used the RMS error, which we defined as:Fig. 5.
As expected, the approximation performs less well when the field to be propagated contains cores of different sizes though the difference in RMS error between the two cases decreases for increasing θ. Even so, as the cross sections in Fig. 4 show, Eq. (5) still leads to an acceptable approximation especially if we take into account that the standard deviation on the core diameter used in our calculations here, is larger than the one observed in SEM images of CFBs fabricated according to our design . Moreover, the computation time with the approximated R-SDI was about 30000 times smaller than that with the exact R-SDI. This was to be expected since with the approximated R-SDI the object field to propagate is (1281 × 761)/19 ≈50000 times smaller. The authors would like to stress that the relative decrease in computation time mentioned here is case specific and depends entirely on the ratio (N × N)/Nc. For example if the object field for the same number of cores Nc is now defined with instead of (N × N) then the relative decrease in computation time will be 4 times smaller. It should also be noted that the decrease in computation time is solely the result of the approximation and no efforts were made to optimize the code for speed.
3. Influence of amplitude and phase noise on the PSF
Using the simplified R-SDI, we computed the PSF for an ideal object field exiting the distal end of our CFB which consists of 25 rings of identical, circular cores on a perfect hexagonal lattice (for a total of 1951 cores) and this for FOV half-angles from θ = 0° up to and including θ = 30° (by adding, on top of a spherical wave front for focusing, the linear phase shift corresponding with θ, as illustrated in Fig. 1 in ). The image plane, with dimensions 394 × 127 μm2 (along the x-axis and y-axis respectively), was located at a propagation distance of 500μm, and centered at (197μm,0μm) to allow the evaluation of the PSF for the different half-angles θ with the same image plane. The resulting PSFs of this ideal field were then used as a reference to compare the PSFs of ‘noisy’ object fields with. We analyzed the influence of two types of object field noise namely noise on the amplitude and phase of the field from each.
First, we characterized the influence of ‘noise’ on the amplitudes of the field coming from each core. Ideally, each single-mode core would emit a field with the same amplitude as long as, for a given length of CFB, the correct input field is coupled into the CFB. However, changes in the thermobaric conditions of the CFB’s surroundings, bending of the CFB and variations in n(x,y) can all lead to unforeseen intercore coupling resulting in variations in the amplitude of the cores’ field in the object plane. To check how robust the PSF would be in the presence of such amplitude variations, we superimposed, for each viewing angle, Gaussian noise with different standard deviations onto the amplitude of the aforementioned ideal field and then recalculated the PSF of this noisy field. The Probability Density Functions (PDFs) used for the Gaussian noise on the amplitude are shown in Fig. 6. Note that as the standard deviation for the amplitude grows, the probability for a negative amplitude grows larger. Since a negative amplitude is equal to a positive one with a π-phase shift, allowing negative amplitudes would also take noise on the wave front into account and therefore we equated all negative amplitudes to zero. This allowed us to also take into account cores that are broken (e.g. fractured or severed) during fabrication or due to careless handling and which can no longer guide light from the proximal to the distal end.
In Fig. 7, the PSFs at θ = 0°, θ = 15° and θ = 30° for σΑ = 0 (reference PSF), σΑ = 0.44 and σΑ = 1 are shown (the PSFs for θ = 5°, 10°, 20° and 25° were also calculated but are not shown for the sake of conciseness). Noticeable is that as the viewing angle θ increases, the amount of speckle like patterns in the image plane is larger for the same σΑ. The cross sections, through the focus, along the x-axis and the y-axis for different values of σΑ and θ are shown in Fig. 8 where we see that the width of the central peak of the PSF doesn’t change much as function of σΑ making the FWHM not useful as the measure of quality. Moreover, the FWHM doesn’t take into account the increase of speckle-like patterns in the background of the image plane (and thus the decrease in signal-to-noise) as can be seen on Fig. 7. Therefore, we opted to use the RMS error of the noisy PSFs with respect to the noise-free PSF as the measure of quality. The RMS error as function of σA at all the angles θ is shown in Fig. 9. For σA< 0.4, the RMS error increases in a near linear way with the slope determined by θ. As σA keeps increasing, the slope of each curve seems to flatten out indicating that for very large σA, the RMS error would remain constant. This is to be expected; as σA increases the PDFs from Fig. 6 go from a Gaussian distribution towards a uniform distribution. Also, for θ = 30° and σA = 1 the RMS error is limited (0.024) which allows us to conclude that the noise on the amplitude of the object field will only be of minor influence on the PSF (as is evidenced by the PSFs shown in Figs. 7 and their corresponding cross sections in Fig. 8).
In a similar way we looked at how noise on the phase of each core would influence the PSF in the image plane. In the noise free case, the phase of the cores was obtained by sampling the noise-free wave front which is the sum of a spherical and a linear wave front. We then added to the phase of each core noise according to a Gaussian PDF (with standard deviation σϕ) and determined, at each of the aforementioned values of θ, the PSF for values of σϕ ranging from 0 to 2π. But as Fig. 10 shows, for σϕ = 2.31 the resulting PSFs are little more than speckle, independent of θ.
This trend can also be seen when we look at the RMS error as function of σϕ (shown in Fig. 11) where we notice that from σϕ = 2.31 onwards the RMS error, for each θ, reaches a plateau around which it slightly oscillates meaning that from σϕ = 2.31 on the coherence of the original wave front is completely lost and that the resulting image will be noise dictated by the random noise of the object wave front. However, it should be noted that σϕ = 2.31 is an overestimation of the wave front noise which will be present in reality. From the datasheet of the Hamamatsu LCOS-SLM X10468 (a spatial light modulator used in ) we can estimate the phase noise for a pixel to be approximately 0.11. For phase noise with σϕ ≤0.11, the RMS error as function of σϕ is nearly linear (Fig. 11 inset left) and the PSFs of the noisy wave fronts and their respective cross-sections (shown in Fig. 12), are almost identical to the noise-free PSF.
In general, the PSF does seem to be surprisingly robust to phase noise. Even for σϕ = π/2, the central peak can be discerned for all angles up to and including θ = 30° as shown in Fig. 13. Also for σϕ in the [0, π/2] range the RMS error increases monotonically with σϕ, with the rate of increase proportional to θ (right inset of Fig. 11). Even so, uncontrolled bending of the CFB can lead to large changes, at the distal end, in both amplitude and phase of the cores with respect to the field in the unbent case. Since we found that even for σΑ = 1 the PSF is almost not affected, we expected that in the presence of both amplitude and phase noise the effect of the phase noise on the PSF will be dominant. To test if this is the case we propagated two distal fields which contained both amplitude and phase noise. The first distal field contained a lot of amplitude noise (σA = 1) but relatively little phase noise (σϕ = 0.11) while the second distal field contained both a lot of phase noise (σϕ = π/2) and amplitude noise (σA = 1). The cross sections of the PSF resulting from the propagation of a field with both amplitude and phase noise are shown in Fig. 14 which shows that when there is a lot of amplitude noise but little phase noise, the resulting PSF closely matches the noise-free PSF (for θ = 0°, 15°, 30° the RMS is 0.0136, 0.0151 and 0.0237 respectively). However, when there is a lot of phase noise as well, the resulting PSF deviates a lot more from the noise free PSF (RMS = 0.0783, 0.0966, 0.1248 for θ = 0°, 15°, 30°) and it more closely matches the PSF of Fig. 13 in which the field to be propagated contained a lot of phase noise, but no amplitude noise.
This allows us to conclude that in case of bending (which causes noise on the amplitude and phase of the distal output field), the proximal input field should be adapted with the emphasis on the compensation of the wave front as bending can cause large phase jumps (>π)  causing the phase relationship between cores to deteriorate which is detrimental for the PSF as evidenced by Fig. 10. Appropriate compensation of the proximal input field requires the knowledge of the magnitude and direction of the bending to which the CFB is subject. This could be achieved for example by integrating the CFB with a 3-core fiber optic shape sensor  into a common catheter allowing for a real-time knowledge of the catheter’s (and thus the CFB’s) shape
In this paper we applied a simplified Rayleigh-Sommerfeld propagation formula for the CFB based on the low diameter-to-wavelength ratio (0.647) of the CFB’s cores. The approximation simplifies the Rayleigh-Sommerfeld diffraction integral and allows a drastic reduction of the computation time without too much loss in accuracy. If the object field consists of Nc cores and is represented by (N × N) points (for adequate spatial sampling of the object field), then the reduction in computing time with the approximation will be in the order of (N × N)/Nc. We then used this approximation to propagate an ideal output field exiting the CFB and determined the PSF for FOV viewing angles ranging from θ = 0° to θ = 30°. If the actual input field differs from the required input field, or if the thermobaric surroundings or the shape of the CFB changes, then the amplitude and phase of the actual distal output field will differ from those of the ideal output field. We modeled this difference as Gaussian noise on the amplitude or phase of the ideal output field. Using different standard deviations we propagated these noisy output fields to determine the corresponding PSF (for the same values of θ) and compared it with the PSF of a noise-free propagated output field. When there is noise on the amplitude only, we found that the presence of the amplitude noise mostly affects the side lobes of the PSF while the central peak, even for considerable amounts of noise (σA = 1), remains relatively unchanged (RMS error≤0.024) for all angles up to and including θ = 30°. When there is phase noise, we found that for σϕ>2.31 the propagated field deteriorates into speckle with no discernable central peak. For noise with σϕ≤π/2 the central peak can still be discerned at all angles up to θ = 30° (as was the case for amplitude noise with σA ≤1), but the corresponding RMS error is higher (0.13 for σϕ = π/2 at θ = 30°) than in the case with noise on the amplitude only. To determine which type of noise influences the PSF the most, we propagated a field which was subject to both amplitude and phase noise. In the case of high amplitude noise (σA = 1) and low phase noise (σϕ = 0.11), the resulting PSF closely matched the noise-free PSF (for θ = 0°, 15°, 30° the RMS is 0.0136, 0.0151 and 0.0237 respectively). However, when the phase noise was high as well (σϕ = π/2), the resulting PSF deviated a lot more from the noise free PSF (RMS = 0.0783, 0.0966, 0.1248) and closely resembled the PSF resulting from the propagation of a field with phase noise (σϕ = π/2) only. This allows us to conclude that phase noise in the CFB’s distal output field is more detrimental to the quality of the PSF then amplitude noise and its compensation should be the main goal during adjustment of the proximal input field.
This research was funded by Stefaan Heyvaert’s Ph.D. grant of the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT- Vlaanderen). R. Buczynski and I. Kujawa were supported by the project operating within the Foundation for Polish Science Team Programme, co-financed by the European Regional Development Fund, Operational Program Innovative Economy 2007-2013. This work was also supported in part by FWO, the 7th FP European Network of Excellence on Biophotonics Photonics 4 Life, the MP1205 COST Action, the Methusalem and Hercules foundations and the OZR of the Vrije Universiteit Brussel (VUB).
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