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Self-reconstruction refers to an ability of certain fully coherent optical beams to recover their spatial profiles after scattering by obstacles. In this communication, we extend the self-reconstruction concept to partially coherent beams. We show theoretically and verify experimentally that any partially coherent beam can self-reconstruct its intensity profile and state of polarization upon scattering from an opaque obstacle provided the beam coherence area is reduced well below the obstacle area. We stress that our self-reconstruction technique is independent of the obstacle shape and it is scalable to the case of multiple obstacles or even of inhomogeneous media as long as a characteristic obstacle area or a medium inhomogeneity scale is well in excess of the beam coherence area or length, respectively. We anticipate the technique to be instrumental in applications ranging from beam shaping to image transfer and trapped particle manipulation in turbid media.
© 2016 Optical Society of America
1. Introduction
It has been known for some time that whenever certain types of fully coherent light beams encounter an obstacle, either in free space or in inhomogeneous media, they are able to reconstitute their spatial shape upon interaction with the obstacle. Optical self-reconstruction has already found applications to microscopic particle manipulation [1], human tissue microscopy [2], quantum entanglement propagation in the presence of obstructions [3] and optical communications through inhomogeneous media [4] among others and it has even inspired science fiction writers.
This self-reconstruction property has first emerged as a signature of diffraction-free beams such as fundamental [5], higher-order vortex [6], fractional [7] and white-light [8] Bessel beams as well as Airy [4, 9] and caustic [10] beams. It was subsequently shown that even some diffracting coherent beams, including optical ring lattices [11], Pearcey beams [12], tightly focused [13] and radially polarized [14] Bessel-Gauss beams, can self-reconstruct. However, the self-reconstruction power of all fully coherent self-healing light beams known to date hinges on engineering special beam profiles and in many cases, it is sensitive to the obstacle size, shape and distribution, thereby limiting applications of such beams.
Partially coherent beams are distinguished by the profound dependence of their spatial profile, polarization, propagation through and interaction with material media on beam coherence properties [15]. In particular, spatial coherence has proven to be a fundamental degree of freedom, indispensable for controlled generation of partially coherent beams with novel properties such as self-splitting [16], flat-top intensity profile [17], radial polarization [18] as well as far-field optical lattice patterns [19, 20], tailored to desired applications, including non-interferometric phase imaging [21], diffractive imaging [22, 23], ghost imaging [24, 25], coherence holography [26], and photovoltaics [27]. The exploration of spatial coherence degree of freedom has also expanded research horizons in nonlinear optics [28–31] and near-field optics [32–36]. Furthermore, optical coherence has emerged as one of the key subjects in recent studies of analogies between quantum and classical entanglement [37–41], and it is anticipated to play a major role in delineating an elusive boundary between classical and quantum realms. The fundamental importance of spatial coherence in classical and quantum physics and the prominent role played by optical self-reconstruction in many promising applications prompt an intriguing question: Can one manipulate spatial coherence of a light source to ensure self-reconstruction of arbitrarily shaped generated beams, scattered by an obstruction or by an inhomogeneous medium, either deterministic or random?
We address this challenge here by introducing a novel self-reconstruction protocol applicable to any nearly incoherent optical beam regardless of its intensity profile and polarization state particulars. We show both theoretically and experimentally that making the spatial coherence area of any optical beam much smaller than the obstacle area is a key prerequisite to successful self-reconstruction of the beam intensity and polarization distributions.
2. Qualitative picture of nearly incoherent beam self-reconstruction
To gain insight into the new protocol, we first consider qualitatively the scattering of a nearly incoherent beam of diameter σI and a characteristic coherence length σc by a circular aperture of diameter σa serving as an isolated obstacle. We assume that σc ≪ σa < σI. The situation is illustrated in Fig. 1. The nearly incoherent beam, which is known to be composed of a number of randomly distributed uncorrelated beamlets of the size σc ≪ σI [15], has a diffraction angular spread θd ∼ λ/σc, entirely determined by the beam wavelength λ and its coherence length σc. This is because each beamlet behaves as a fully coherent structure of the size equal to the overall beam coherence length. Since the obstacle causes the scattering angular spread of θs ∼ λ/σa ≪ θd, the nearly incoherent beam is relatively impervious to the obstacle and is able to self-reconstruct its intensity profile and polarization state upon scattering from it.
Fig. 1 Qualitative illustration of nearly incoherent beam scattering by an obstacle. Nearly incoherent beam of diameter σI and coherence length σc has a specular structure with each speckle beamlet having the spot size equal to the beam coherence length. The obstacle is modeled by a circular aperture of width σa.
The presented self-reconstruction principle can be naturally extended to the case of multiple obstructions such that the smaller of a characteristic obstacle scale or an inter-obstacle distance is still much greater than the incident beam coherence length. In case of a continuous inhomogeneous medium, the present self-reconstruction scenario can work out provided a characteristic refractive index inhomogeneity scale of the medium dwarfs the beam coherence length.
The outlined self-reconstruction principle is clearly independent of the obstacle shape and the necessary condition for beam self-reconstruction can be formulated in terms of the beam coherence area Ac, the obstacle area Aa and the beam intensity cross-section area AI as
We stress here a fundamental difference of fully from that of partially coherent beam self-reconstruction. While the former relies on special properties of diffraction-free complex amplitude profiles, the latter is ensured by judicial beam coherence engineering.3. Theoretical formulation and numerical self-reconstruction
To confirm quantitatively the just described qualitative picture of partially coherent beam self-reconstruction, we introduce the cross-spectral density tensor of the electromagnetic field E ensemble, describing random beams in the source plane z = 0, as [15]
Here i, j = x, y;ρ is a position vector in the plane transverse to the beam propagation direction, and the angle brackets denote statistical ensemble averaging. We specifically consider the input beam of a Schell-model type such that where Ii,j stand for intensity components along the x, y-axes and gi,j describes (statistically homogeneous) second-order field correlations. A Schell-model beam can be readily generated by transmitting an arbitrary coherent beam through a rotating ground-glass disk [32]. The transmitted beam coherence length can be controlled by adjusting the disk rotation speed and the incident beam spot size at the ground-glass disk.Next, we place an obstacle illuminated by the Schell-model beam in the focal plane behind a thin lens with the focal length f = 250mm (source plane) and examine the scattered beam image in the focal plane in front of the lens (image plane). The lens performs a Fourier transform of the scattered partially coherent beam cross-spectral density tensor [33] (see Appendix A). The intensity and polarization state of the scattered beam can be inferred from the Stokes parameters defined as [34]
here S0(ρ) represents the average beam intensity.As a numerical example, we studied the scattering of a radially polarized Schell-model (RPSM) beam introduced in [18] (see Appendix A for beam structure details) with the beam width σI = 1.5mm and coherence length σc = 75µm by a sector-shaped opaque object (SSOO) with a center angle φ and diameter σa = 1.5mm. We exhibit the simulation results in Fig. 2. The left column, Figs. 2(a), 2(d), 2(g), 2(j), and 2(m), shows the unobstructed beam intensity in the source plane and in the image plane and polarization distributions in the image plane, respectively, while the middle one, Figs. 2(b), 2(e), 2(h), 2(k), and 2(n), demonstrates the beam self-reconstruction in the presence of the SSOO blocking most of the input beam cross-section. The SSOO area conforms to the constraint set by Eq.(1). We also show in the right column of Fig. 2 that as the SSOO transmittance area approaches the beam coherence area, violating the conditions of Eq.(1), the beam self-reconstruction ability is severely inhibited.
Fig. 2 Numerical self-reconstruction of a radially polarized Schell-model beam scattered by a sector-shaped opaque object (SSOO). Intensity and Stokes’ parameter distribution of the beam in the source plane and in the image plane with no obstruction (left column) and with the SSOO with the angle φ = 7π/4 (middle column) and φ = 71π/36 (right column). The beam parameters are: σI = 1.5mm and σc = 75µm and the SSOO has the same diameter as the beam, σa = σI.
We note that (nearly perfect) self-reconstruction of the beam is realized in the image plane of the lens. There are some distortions caused by the obstacle immediately after scattering and on propagation to the image plane. We exhibit the RPSM beam transient dynamics at several distances behind the image plane in Figs. 3 and 4, and its self-reconstruction ability in Figs. 5 and 6.
Fig. 3 Numerical transient dynamics of the RPSM beam scattered by a sector-shaped opaque object (SSOO) at several distances. a Intensity distribution of the beam at several propagation distance without obstructed. b Intensity distribution of the beam at several propagation distance with the SSOO center angle φ = 7π/4 and c with the SSOO center angle φ = 71π/36. The beam parameters are the same as Fig. 2.
Fig. 4 Experimental transient dynamics of the RPSM beam scattered by a sector-shaped opaque object (SSOO) at several distances. a Intensity distribution of the beam at several propagation distance without obstructed. b Intensity distribution of the beam at several propagation distance with the SSOO center angle φ = 7π/4 and c with the SSOO center angle φ = 71π/36. The beam parameters are the same as Fig. 2.
Fig. 5 Numerical self-reconstruction ability of a RPSM beam scattered by a SSOO. a Intensity distribution of the beam in the source plane with SSOO center angle φ = 71π/36. b, c, d and e Intensity distribution of the beam in the focal plane with different coherence length σc.
Fig. 6 Experimental self-reconstruction ability of a RPSM beam scattered by a SSOO with center angle φ = 71π/36. a, b, c, and d Intensity distribution of the beam in the focal plane with different coherence length σc.
4. Experimental results
4.1. Case I: Single obstacle
We carried out experimental self-reconstruction of an RPSM beam upon its scattering from a single sector-shaped opaque object. The RPSM beam and the imaging lens had the same parameters as in our numerical simulations. The experimental setup, sketched in Fig. 7, was identical with that in [18] (see Appendix B for particulars) except the SSOO, placed in the focal plane of lens L3, was present. In particular, a fully coherent beam, generated by a He-Ne laser was transmitted through a beam expander (BE) and having been reflected by a reflecting mirror (RM), the beam passed through a radial polarization converter (RPC). We then converted a fully coherent radially polarized beam into the RPSM by transmitting the former through a rotating ground-glass disk (RGGD) and Gaussian amplitude filter (GAF). The resulting RPSM beam was scattered by an SSOO with the variable center angle φ, placed in the source plane of the lens L3, optically Fourier transformed into the lens image plane and registered by a charge-coupled device (CCD).
Fig. 7 Experimental setup for RPSM beam self-reconstruction demonstration. Beam expander (BE), reflecting mirror; (RM), radial polarization converter (RPC), rotating ground-glass disk (RGGD); L1, L2, L3, thin lenses, Gaussian amplitude filter (GAF), sector-shaped opaque obstacle (SSOO), charge-coupled device (CCD).
We display the experimental results in Fig. 8 where the left column shows the unobstructed RPSM beam intensity and polarization distributions in the source and image planes. In the middle column, we display the same quantities when the beam is scattered by the SSOO that covers half the cross-section area of the RPSM beam in the source plane. The accurate beam intensity profile and the polarization state recovery is evident from the figure. The right column of the figure exhibits high-quality self-reconstruction of the beam subject to the SSOO obstructing the greater part of the beam cross-section. The RPSM beam is linearly polarized with the polarization direction making the angle θ with the x-axis. The RPSM beam polarization direction in the image plane is indicated by the arrow.
Fig. 8 Experimental beam self-reconstruction: Case 1. SSOO. Left column: Intensity and polarization distributions of the RPSM beam in the source and focal planes in the absence of obstruction. Middle column: The same as in the left column, except the SSOO blocking half of the beam cross-section is placed in the source plane. Right column: Same as the middle one with the SSOO having the center angle φ = 7π/4. The linearly polarized-with an inserted linear polarizer not shown in Fig. 3-imaged beam has the polarization direction indicated by the arrow. The beam polarization direction makes the angle θ with the x-axis.
4.2. Case II: Scattering by turbid medium
We also examined self-reconstruction of the RPSM beam transmitted through a colloidal medium. A glass of diluted milk, shown in Fig. 9 (c), served as the scattering medium in our experiments. The medium can be viewed as random because the colloidal particle density fluctuations in the milk lead to the refractive index fluctuations. The latter cause scattering of the light transmitted through the glass. In Figs. 9(a) and 9(b) we display schematics of the experimental setup for the cases of fully coherent radially polarized and RPSM beams, respectively. The RPSM beam coherence length and beam width were taken to be σc = 75µm and σI = 1.5mm, respectively. The beam width of fully coherent RP beam was taken to be σI = 350µm to make sure that the beam size in the detect plane was nearly same with that of the focusing RPSM beam in the detect plane. Each beam was transmitted through the glass of milk and then focused by a thin lens L with the focal length f = 250mm onto the CCD camera located in the image plane of the lens. In Figs. 9(d) and 9(e) we exhibit the experimental results of the intensity profile of the transmitted radially polarized coherent beam and that of the RPSM beam at different time instances. We can clearly detect the difference in the quality of self-reconstructed images. While the coherent transmitted beam is strongly affected by the medium resulting in poor self-reconstruction with a granular structure, the transmitted RPSM beam is seen to possess a smooth spatial intensity distribution, reproducing well the RPSM beam in the image plane.
Fig. 9 Experimental beam self-reconstruction: case 2. Turbid medium. (a) Experimental setup for analyzing fully coherent radially polarized beam transmission through a turbid medium with random refractive index fluctuations. (b) Experimental setup for analyzing the RPSM beam transmission through the same turbid medium. (c) Photograph of the turbid medium (diluted milk). (d) Experimental intensity profile self-reconstruction of the fully coherent radially polarized beam passing through diluted milk at different instances of time t. (see also Visualization 1) (e) Experimental intensity profile self-reconstruction of the RPSM beam passing through diluted milk at different time instances t. (see Visualization 2 as well).
5. Promising applications and summary
Apart from the image and/or information transfer and optical communications through adverse inhomogeneous environments, the proposed self-reconstruction protocol can be useful for generating novel types of partially coherent sources, inaccessible in the realm of coherent optics. The latter property stems from the inherent ability of sources with controlled spatial coherence to self-reconstruct whenever the source cross-section area is partially obstructed by an obstacle provided the source coherence area is engineered to be much smaller than the obstructed source area. This opens up an exciting possibility for realizing periodic arrays of partially coherent beams with prescribed spatial intensity profile and state of polarization in the focal plane of a lens by focusing vector partially coherent beams with lens arrays. As an illustrative example, we consider focusing of an RPSM beam with the same parameters as in the previous sections by a 3 × 3 thin lens array with the focal length f = 65.6mm. The schematic of the experiment is shown in Fig. 10(a). Both the diameter of each lens and the distance between the centers of adjacent lenses were taken to be 1mm. Due to the RPSM beam self-reconstruction upon scattering by each individual lens aperture, the transmitted field from each lens forms an annular beam spot with radial polarization in the focal plane of the lens. Consequently, a 3 × 3 radially polarized beam array is formed in the focal plane. In Figs. 10(b)–10(e), we display our simulation results while Figs. 10 (f)–10(i) show an experimentally generated radially polarized beam array. To emphasize the spatial coherence role in the microlens array formation, we show how the array forms as a function of the input beam coherence in Fig. 11 and in Visualization 3. We anticipate these microlens arrays to find applications for multiple particle trapping and manipulations. Next, thereby synthesized microlens arrays can be used in conjunction with the Shack-Hartmann wavefront sensing protocol to determine the coherence state of a test source beam via the wavefront sensing [45,46]. Furthermore, we point out an attractive extension of our protocol to the case of disordered self-focusing nonlinear media. The self-focusing nonlinearities in such media will oppose rapid diffraction of low-coherence self-reconstructed beams and open up an opportunity to transmit the input beam images over longer distances than it is possible in random linear media. Finally, the present self-reconstruction should be distinguished from partially coherent beam self-imaging that only occurs at certain distances within graded-index media [47].
Fig. 10 Radially polarized beam array generation through the RPSM source self-reconstruction. (a) Schematics of the apparatus for generation of a radially polarized beam array. (b) Numerical simulation results of the generated beam array intensity profile and the state of polarization. Both the diameter of each lens and the distance between the centers of adjacent lenses are 1mm. The focal length of the lens array is 65.6mm. (c), (d), (e) Stokes’ parameter distributions of the radially polarized beam array; (f), (g), (h), (i) experimental results of the generated radially polarized beam array intensity and the state of polarization. The arrow indicates the polarization direction of the beam.
Fig. 11 Numerical formation of radial polarized beam array as a function of the input beam coherence. a-e Numerical simulation results of the generated beam array intensity profile with difference beam coherence.
In summary, we have shown that any partially coherent beam, scattered by an arbitrary opaque obstacle, can recover its intensity profile and the state of polarization provided the spatial coherence area of the beam is made much smaller than the obstacle area. The presented self-reconstruction technique is scalable to the case of multiple obstacles with a characteristic inter-obstacle distance being much greater than the obstacle size. The technique also extends to scattering in inhomogeneous media, either deterministic or random, with a characteristic inhomogeneity scale dwarfing the beam coherence length.
Appendix A. Numerical simulations
In our numerical simulations and experimental demonstration of self-reconstruction, we employ radially polarized Schell-model (RPSM) beam which was introduced and studied in detail in [18]. Our RPSM beams have Gaussian intensity profiles with the same width σI = 1.5mm along each Cartesian axis and the two-point correlation tensor gij with the diagonal and off-diagonal components in the form
where a = x, y and respectively. The correlation length is σc = 75µm.The opaque obstacle is modeled by a generic transmittance amplitude T (ρ) such that
The cross-spectral density tensor of the RPSM beam immediately behind the obstacle reads
An imaging lens with the focal lens f = 250mm transforms the scattered RPSM beam, placed in its focus, into an image in the focal plane in the front of the lens. Thus, the lens performs a Fourier transformation of the scattered cross-spectral density tensor yielding
where tilde denotes a Fourier transform. Using Eqs. (10)–(12) as well as the Stokes parameter definitions, Eqs. (4) to (7) in the main text, we can determine the intensity and polarization distribution of the imaged RPSM beams.Appendix B. Experimental realization
The 632.991nm carrier wavelength He-Ne Laser used in our experiment is fabricated by Thorlabs, Inc. (HRS015B). The beam expander is manufactured by Daheng Optics, Inc. (GCO-2501). It has an operating wavelength of 450nm to 680nm, and expanding ratio of 5 to 10. The radial polarization converter is used to convert the linear polarization to radial polarization (RADPOL4, Arcoptix). The surface roughness of ground-glass disc is 400 mesh number and the rotating speed is 3000r/min. The CCD camera is acquired from Point Grey, Inc. (GRAS-20S4M-C) with its resolution of 1624 × 1224, and Frame Rate, 30 FPS. The integration time of CCD camera during optimization was 20ms and the Frame rate was set to 10 FPS for video. The 3 × 3 lens array (APO-Q-P1000-R30) is acquired from Advanced Mirooptic Systems Gmbh and its focal length f = 65.6mm. Both the diameter of each lens and the distance between the centers of adjacent lenses are 1mm.
Funding
National Natural Science Fund for Distinguished Young Scholar (11525418); National Natural Science Foundation of China (11274005, 11474213); Project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions; National Science and Engineering Research Council of Canada (NSERC);Innovation Plan for Graduate Students in the Universities of Jiangsu Province (KYZZ16_0079).
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