Exploiting scattering media for exploring 3D objects

Transcription

1 Exploiting scattering media for exploring 3D objects Alok Kumar Singh 1, Dinesh N Naik 1,, Giancarlo Pedrini 1, Mitsuo Takeda 1, 3 and Wolfgang Osten 1 1 Institut für Technische Optik and Stuttgart Research Center of Photonic Engineering (SCoPE), University of Stuttgart, Pfaffenwaldring 9, Stuttgart, Germany School of Physics, University of Hyderabad, Hyderabad , India 3 Center for Optical Research and Education (CORE), Utsunomiya University Yoto 7-1-, Utsunomiya, Tochigi, , Japan 1

2 Experimental Setup The sketch of the experimental setup of looking through a diffusing media is shown in Figure 1 of the main article. Here the schematic diagram is shown with more details in supplementary Figure S1a. The light from a Nd:YAG laser with 53 nm was divided into two parts. One part was coupled into a pig-tail fiber (not shown in the figure) and served as an off-axis reference point source. The other part was expanded with a f 0 mm focal length lens (L1) and was made to pass through a rotating ground glass (RD), which serves as a spatially incoherent extended source for object illumination. A transmissive object, 1951 USAF test target was placed at a distance 40 mm in front of the beam splitter (BS); the object was made coplanar to the reference point source for D imaging. Similarly for imaging using a thick chicken breast sample and around the corner a diffused object composed of a 3 mm high and mm wide transmissive letter H pasted on a thin ground glass, was imaged with lens L ( f 60 mm ) to form a demagnified aerial image in front of the diffuser at distance 40 mm in front of it. Whereas for 3D imaging another object with a triangular and rectangular window was created. The axial and lateral separation between the two windows were mm and 6 mm respectively, and one of the objects were coplanar to the reference. The diffusing medium was kept at the same distances zˆ z 600 mm from the reference and observation planes, so that shift-invariance was realized through unit magnification. As explained in the principle, light beams from the object and the reference are mutually incoherent and do not interfere with each other to form a hologram on the diffusing surface. Instead, both pass through the diffuser independently and create own speckle patterns, which are superimposed incoherently on the observation plane and detected with a CCD image sensor. The intensity interference between the independently created speckles was achieved numerically through the computation of

3 the cross-(auto)-covariance. An aperture attached to the diffuser controls the speckle size and the depth of field (DOF) of the imaging system. A similar schematic of the optical geometry for looking around the corner is shown in Supplementary Figure S1b. Here the transmissive diffusing medium was replaced with a rough aluminum plate with a thin but strong scattering layer on its surface. To avoid specularly reflected light, the center of CCD was kept in the line normal to the illuminated area on the scattering surface. The red arrow on the surface indicates in the direction normal to the surface. 3

4 Supplementary Figure S1: The schematic diagrams. (a) Imaging using a diffusing medium and (b) looking around the corner. The object and the reference are kept in such a way that the condition of shift-invariance is satisfied and the speckles produced by both, after propagating through the diffuser, are correlated. L1: focusing lens, RD: rotating diffuser, M1 and M: mirrors, L: imaging lens and BS: beam splitter. 4

5 3D imaging through a diffusing media Supplementary Figure S: Impulse response of the diffusing medium; A point source is placed in the reference plane to observe the behavior of the diffusing medium in the observation plane. The speckle pattern created in the observation plane gives a point spread function (PSF) of the diffuser, which is shown to have 3D shift invariance when the geometry satisfies certain conditions. To realize the goal of 3D imaging through a diffusing medium, the 3D memory effect (i.e. the shift invariance of the point spread function (PSF) for both lateral and axial coordinates) of the diffuser has been exploited in the main text without proof. Here we show that PSF can have the 3D memory effect for a thin diffusing medium, and clarify the conditions for this to be true. Referring to Supplementary Figure S, we first consider the field impulse response of the diffusing medium defined as the complex field h( r, r ; z, z ) created (at point r in the observation plane at distance z from the diffuser) by a point source at r in the reference plane at distance z from the diffuser. The field at point r in front of the diffuser can be written as 5

6 exp( ikzˆ ) ˆ G r r ( r, rˆ ; zˆ ) exp ik izˆ zˆ, (S1) The field immediately behind the diffuser is given by multiplying the pupil function of the diffusing media P( r ) P( r ) exp[ i R ( r )] with the field in front of the diffuser, where R ( r ) is the random phase introduced by the thin diffuser. Thus the scattered field on the observation plane is given by h( r, rˆ ; z, zˆ ) G ( r, r ; z) P ( r ) G ( r, rˆ ; zˆ ) d r, (S) where G( r, r ; z) is a kernel for Fresnel diffraction; exp( ikz) r r G( r, r ; z) exp ik. (S3) iz z After substituting for G ( r, rˆ ; zˆ ) and G( r, r ; z ) from equations (S1) and (S3) into equation (S), we have the field impulse response in the observation plane: exp( ikzˆ ) exp( ikz) r rˆ r r h( r, rˆ ; z, zˆ ) P( ) exp ik exp ik d izˆ iz r r zˆ z. (S4) ik rˆ r ik 1 1 rˆ r exp P( )exp d zˆ z r zˆ r z zˆ r z r The intensity of the impulse response, namely PSF, of the diffuser is thus given by ik ˆ (, ˆ;, ˆ) (, ˆ 1 1 r r S r r z z h r r; z, zˆ ) P( r )exp d zˆ z r zˆ z r r. (S5) 6

7 In analogy to the imaging condition of a thin lens, we introduce a new variable f as a focal length of the lens (which we call Freund s wall lens), such that (S6) ẑ z f To examine the shift invariance (or memory effect) of PSF, let us introduce small shifts to the lateral and longitudinal coordinates: S( r r, rˆ rˆ ; z z, zˆ zˆ ) ik 1 1 rˆ rˆ r r P( r )exp d zˆ zˆ r z z zˆ zˆ r z z r. (S7) In order for the PSF to have 3D shift invariance S( r, rˆ ; z, zˆ ) S( r r, rˆ r ˆ; z z, zˆ zˆ ), we set the following conditions on equation (S5) and equation (S7) zˆ z 1 zˆ z f zˆ z zˆ zˆ z z zˆ z zˆ z f zˆ z (S8) rˆ r rˆ rˆ r r rˆ r rˆ r zˆ z zˆ zˆ z z zˆ z zˆ z, (S9) where the approximations have been made based on the assumption of paraxial optics and small shifts. From equation (S9) and equation (S8), we have z zˆ ˆ mˆ r r r (S1 0) z z zˆ zˆ m zˆ (S11) 7

8 where m z zˆ and m ( z zˆ ) are, respectively, the lateral and axial magnifications for the imaging condition equation (S6) by the Freund wall lens. Thus we have shown that the PSF of the diffuser can be made to have a 3D memory effect by properly rescaling the coordinates in accordance with equation (S10) and equation (S11). For the special case in which z zˆ and m 1, the PSF can take the form of a convolution kernel S( r, rˆ ; z, zˆ ) S( r r ˆ; z zˆ ) with a perfect 3D memory effect with no need of rescaling. Resolution of 3D imaging through diffusing medium Here we show that the lateral and axial resolutions of the proposed scheme for 3D imaging through a diffusing medium are same as those of conventional diffraction-limited imaging optics. As seen in equation (1) in the main text, the resolution is determined by the cross-covariance of the PSF [ DS( r; Dz) ] Ä[ D S( r;0) ], which we have approximated by a delta function for simplicity of explanation. We evaluate the cross-covariance of the PSF from the cross-correlation of the field impulse response h( r, r ˆ; z, zˆ ). From the central limit theorem, we assume that the field scattered from the diffuser obeys circular Gaussian statistics with zero mean, so that the cross-covariance of the PSF is reduced to the squared modulus of the cross-correlation of the field impulse response 1, [ ] [ ] DS( r; Dz) Ä D S( r;0) = D S( r + D r, rˆ ; z +Dz, zˆ ) DS( r, rˆ ; z, zˆ ) = h r + D r rˆ z +Dz zˆ h r rˆ z zˆ * (, ;, ) (, ;, ), (S1) where... denotes ensemble average. From equation (S4) we have 8

9 [ ] [ ] * DS ( r; Dz) Ä D S( r;0) = h( r + D r, rˆ ; z + Dz, zˆ ) h ( r, rˆ ; z, zˆ ) = òò ì * ik 1 1 ˆ + D ü P( ') P ( ) expï éæ ö æ ö ù ' r r r r r í ' ï ý êç zˆ z z r è + ø çè zˆ z + zø r ú ï ë û î D D ïþ ì ik éæ1 1 ˆ ù exp ö æ ö r r ü í ï- êç êè ç zˆ z r ø çè zˆ z r ïd ' d ý r r ë ø ú ï û î ïþ. (S13) Assuming that the field exiting from the diffuser pupil is delta-correlated such that * ( ') ( ) = ( ) P r P r P r ' d( r '- r), (S14) we can rewrite equation (S13) as ì ik 1 1 ü P( ) expï éæ ö æ ö ù r +Dr r D r D Ä D r = í ï ò r ýd êçè z + z z r ø èç z + z z r r ø ú ï ë û î D D ïþ [ S( ; z) ] [ S( ; 0) ]» ò ì ik z ü P( ) expï éæ ö æ ö ù D Dr r í- + ï d ý êèç z r ø èç z r r ø ú ï ë ûï î þ. (S15) Note that the cross-covariance of the PSF of the diffuser has the same mathematical form as a local 3D intensity distribution I( r, z) of the diffraction field created near the focal point by a converging spherical wave exiting from the pupil aperture P( r ) with the radius of curvature z. Thus we have shown that the lateral and axial resolutions of the proposed technique are the same as those of an ideal diffraction-limited imaging optics. For a circular aperture with a diameter D, the lateral and axial resolutions are given 3 by dr» l( z D) and dz» l( z D). corr. 1.4 corr. 6.7 Remember that these resolutions are defined in the side of the observation space. The corresponding resolutions in the side of the object space are scaled by the magnifications as corr.» l( ) ( ) and dz ( ) ( corr. 6.7l z D 1 m ) drˆ 1.4 z D 1 m». 9

10 References 1. Goodman JW. Statistical Optics. New York: John Wiley & Sons, 1985, pp Wolf E. Introduction to the Theory of Coherence and Polarization of Light. Cambridge: Cambridge University Press, 007, pp Goodman JW. Speckle Phenomena in Optics: Theory and Applications. Roberts & Company: Englewood, Colorado, 007, pp84. 10