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- Guided wave photonics saunders college publishing electrical engineering
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All these outcomes can be thought of as enhancing a particular mode of the output field. In this letter, we study the relation between the attainable enhancement factor, corresponding to the efficiency of mode conversion, and the field distribution of the target mode. Working in the limit of a thin diffuser enables not only a comparison between experimental and simulated results, but also allows for derivation of an analytic formula.
These results shed light on the ability to use a scattering medium as a mode converter and on the relationship between the desired shape and the efficiency. When coherent light passes through a medium that exhibits an inhomogeneous index of refraction, interference between scattered-light components occurs and is manifested as random speckle patterns with an intensity distribution that appears highly disordered and stochastic. However, these complex random interferences are completely deterministic and can, therefore, be manipulated by controlling the input mode of the field coherently.
The availability of Spatial Light Modulators SLMs has allowed controlling this type of optical fields with many degrees of freedom and opened the way to a class of experiments and applications that deals with complex optical fields [ 1—12 ].
Exploiting the deterministic nature of light scattering, Vellekoop and Mosk [ 1 ] demonstrated coherent control of multiple scattering to generate a focus behind the turbid medium. By adjusting the target function used as feedback, they have shown that not only were they able to enhance a single focal point in the target plane but also multiple foci simultaneously.
In contrast with the more common application of wave front shaping, where the goal is to enhance one speckle grain or more [ 12—15 ], it has been recently shown that adaptive optimization of the incident wave front can also practically eliminate the intensity of light that scatters into some aperture, for example the light that is scattered into a camera, as long as the aperture is small enough [ 16 ].
Enhancing and reducing intensity by wave front shaping through scattering media over a broad area has been also demonstrated in [ 17—19 ], and while this subject is being thoroughly studied, there is also an increasing interest in manipulating the complex speckle field into other spatial modes [ 20,21 ].
One such example is the generation of reconfigurable Bessel-like beams at user defined positions through scattering media by exploiting adaptive focusing [ 22,23 ]. Another example is generating spatial modes prior to the scattering media and then using a control-SLM for sorting the input light modes into different output channels [ 24 ].
While the limits to the efficiency of focusing light into an isolated focal spot are well understood, the question of how effective the conversion of scattered field into some general mode has not been thoroughly explored. Here we generalize the concept and study the enhancement of several desired target fields of light over a region behind a scattering media, so as to examine how the achievable conversion efficiency depends on the distribution of the target field and on its size.
We focus here on target fields with uniform amplitude yet with some desirable phase structure. In order to study a general mode conversion, we choose three types of phase structures that exhibit very different properties from one another: homogenous constant phase , completely inhomogeneous complex phase structure that is assigned according to random distribution and slowly varying spiral phase.
We choose to work at the limit of a thin diffuser, as this allows a detailed comparison between experiments, simulations and analytical or semi-analytical derivations, which in turn provide some intuition relating to the origin of the dependence of the conversion efficiency on the desired mode. Figure 1 a illustrates an experiment that demonstrates the enhancement of a general target field: A telescope expands the laser beam which then reflected off the SLM.
The intensity of the beam on the SLM has a Gaussian distribution; hence each pixel on the SLM contributes differently to the enhancement process. The SLM which is divided into 15X20 effective blocks is then imaged on the diffuser with 5x de-magnification imaging set up. Light scattered by the diffuser creates a speckle pattern that propagates and encounter a phase mask PP at the far field at the Fourier plane of a lens L1.
A variable aperture A attached to a phase mask is positioned at the target plane and it is yet another amplitude degree of freedom to determine the size of the spatial mode we wish to enhance. A speckle field propagates and encounters a helical phase plate PP at the far field or at the Fourier plane of lens L1, as shown. The resulting field is then focused on a CCD camera by lens L2. Using a feedback algorithm, the phases on the SLM are adjusted to maximize the focus, which corresponds to a uniform phase distribution at the output of the phase mask.
A variable size aperture A , attached to the mask, enables the control the size of the target mode. Since we take the FWHM of the distribution as the speckle size, the diameter is smaller than the full width by a factor of about 1.
A doughnut shape which is associated with the helical phase distribution is obtained. A feedback algorithm [ 25, 26 ] is used to find the SLM phase pattern that maximizes the intensity at a target pixel on the CCD, that is, the pattern that maximizes a plane wave at the back surface of the phase mask.
The field at the front surface of the phase mask correspondingly obtains the phase conjugate distribution of the mask. Since the SLM is based on a single liquid crystal device, its operation is limited to a single polarization. Therefore all experiments are performed with a linearly polarized input beam.
Notably, this is not an intrinsic limitation of the method. Figure 1 c shows that after the optimization process, the intensity distribution of the scattered light acquired a doughnut shape which is consistent with the helical phase structure of the spiral phase plate with topological charge of 2 used in the experiment.
We studied both the dependence of the enhancement on the aperture size and, more importantly, on the shape of the desired mode. Figure 2 shows the maximal enhancement factor obtained experimentally as a function of the aperture diameter for three target fields. The enhancement factor is defined as the ratio between the optimized intensity on the target CCD pixel and the initial average intensity before optimization.
Every data point is taken as the average value of 4 different realizations of the scattering medium. The optimization procedure was performed using the MATLAB genetic algorithm toolbox, with the target of increasing the total intensity at a small region typically about the size of a single speckle on the CCD [ 25 ]. The enhancement factor is plotted as a function of aperture diameter for no mask blue helical phase mask of topological charge 2 red and a random phase mask black.
The upper axis indicates the aperture diameter in terms of average speckle diameter. Error bars were determined from multiple realizations of the scattering medium. The plots exhibit the following trends: For a constant phase i. For helical phase, the enhancement factor increases at first, and then decreases with the aperture size. The enhancement factor for a target field with quasi-random phase distribution exhibits an approximately constant value, higher than those obtained for the other phase structures.
As is clearly shown by the experiments, the attainable enhancement strongly depends on the shape of the desired output mode, as well as on the size of the optimized field.
These somewhat surprising results call for a more detailed analysis of the origin of this behavior. For further understanding of the experimental results we performed computer simulations and derived theoretical estimates of the optimal enhancements for the same three types of phase masks.
The simulations are performed by averaging 50 realizations of speckle fields produced by a 64X64 random phase matrix padded with zeros to form a X matrix. This leads in the far field to a speckle field speckles in total, each about 4 cells wide. The maximal enhancement is estimated assuming control elements in the SLM plane, and calculating the optimal control phases by first estimating the phase each SLM pixel contribute to the target field, and applying the conjugate phase to the appropriate control element to assure constructive interference of all contributions.
The enhanced value is compared to the value without control averaged over realizations to extract the enhancement factor. Simulation results are shown in Fig. Quantitatively, the experimentally observed enhancements were lower than the theoretically predicted values.
One possible reason could be the non-uniform Gaussian intensity distribution of the illuminating beam on the SLM. As a result, the control elements do not contribute equally to the enhancement. The derivation of the random phase is not valid for small apertures details in theoretical derivation section.
The enhancement factor is defined as the ratio between the optimized intensity on the target pixel and the initial average intensity before optimization. The experiments utilize thin diffusers. This enables us not only to simulate the system response but also to derive an analytical expression to estimate the expected enhancement.
These estimates are based on Fourier analysis of the propagation of the speckle field, as detailed below, and the results are shown in Fig. As can be seen, the analytical model agrees well with the simulation results. Notably, the model captures the non-trivial dependence of the maximal enhancement on the aperture size for the three different targets, and it can help us to get some insight into this behavior. We wish to control the field shape across an aperture, assumed here circular with diameter S, at a distance L.
The target field is phase conjugated by a phase mask M and the goal is to maximize the DC component that is, the light at the focal point P of a Fourier Transform lens as shown. The distribution of light on the SLM is determined by the convolution of the phase mask and the aperture. For a constant phase, as illustrated here, as the aperture is closed the diffracted beam broadens, covering more pixels on the SLM, enabling higher enhancement.
In the simplest case, we wish to maximize a uniform field over the aperture S that is, no mask. When S is completely open, the system images one point on the scatterer to the focal point, hence only one element of the SLM is effective, and no control is possible. More control is possible for smaller apertures. This is easier to see if we consider a wave propagating backwards from P towards the scatterer, as shown in Fig. To quantify this, consider the diffraction of the backward- propagating wave.
To estimate the maximal possible enhancement we need to integrate the fields arriving from all control elements to P. Without control, these fields arrive with random phases, while proper control will add them all in-phase. We do that by counting the elements included in each ring around the optical axis, weighted by the field at that radius. Without optimization, the intensity is the result of addition of phasors with random phases yielding:.
The optimal enhanced field is obtained by aligning all phasors:. We then get for the enhancement with optimal phases:. This is shown as the red line in Fig. This happens when the aperture is smaller than a speckle at the target plane, and then the optimum is achieved by simply enhancing the intensity in this single speckle, which reproduces the Vellekoop-Mosk factor. Consider now a mask with phase vortex of charge 2, i.
The field on the SLM is now. The enhancement F s is shown as the green line in Fig. Experimental corroboration of this limit is, however, inaccessible, since the intensity at the node of the spiral phase rapidly decreases for small apertures. From these we get the enhancement factor:. Note that in this case, the enhancement is predicted to be independent of the aperture size, as long as the aperture is significantly larger than the phase correlation length of the speckle field, corresponding roughly to the size of a single speckle grain.
It is known that the attainable enhancement depends on the number of control elements that take part in the optimization. As we have discussed above, when the target is to match a particular mode over some area, the number of contributing control elements and their relative contributions depend on the target field and its size.
For the case of a target plane-wave with a flat phase, the field on the SLM is simply the backward-diffraction pattern of the aperture as demonstrated in Fig. Similar convolution-diffraction reasoning have been applied for other modal structures. It is clear, for example, why it is easier to optimize a field with random patterns e. The light from most of the SLM elements reaches the target pixel, regardless the aperture size; therefore, the enhancement factor is nearly constant because the same numbers of control elements are contributing to the optimizations process.
Guided wave photonics saunders college publishing electrical engineering
Acoustical methods are playing an increasingly important role in the measurement of small-scale processes in the ocean. Most of the applications to date are based on backscatter sonar in various forms; and the techniques of Doppler and echo sounder measurement of ocean currents, internal waves, surface waves, bubble fields, mixing processes, and turbulence, as well as biological phenomena, are in a state of rapid development. An alternative and less well developed technology exploits a bistatic system transmitter and receiver separately located to measure the influence of the medium on signals traveling along wholly refracted paths. Properties of the medium are then recovered through an inversion of the detected signals. This, of course, is also the goal of acoustic tomography Munk et al. We refer to this approach as acoustical scintillation. Many reports on these developments have appeared in the acoustic and oceanographic literature using fixed-bottom-mount frames that are cabled to shore.
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Nishizawa, G. Effects of small-scale heterogeneities on seismic waveform fluctuations were studied by physical model experiments. Using a laser Doppler vibrometer, we recorded elastic waves propagating through a granite block at observation points that were arranged as an equally spaced circular array. A disc-shaped PZT source was attached on the other side surface of the circular array for realizing equivalent positions with respect to both source radiation pattern and travel distances of waves. Waveform pairs were selected out from the waveforms, and cross spectra of time-windowed partial waveforms were calculated by applying the multivariate AR model.
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