Jones Matrix Imaging for Transparent and Anisotropic Sample Theory of Jones Matrix: The plane wave components of the optical field can be written as: According to Jones matrix formalism, E’x =Jxx Ex +Jxy Ey E’y =Jyx Ex +Jyy Ey For example the linear polarizer is characterized by the relation: 0 ≤ Px,y ≤ 1

Experimental techniques to determine Jones Matrix Applications of Jones Matrix To understand the light matter interaction To measure the stress and strain in materials In biology - for diagnosis of diseases In cosmetic industries To understand the propagation of light through several polarizing elements Experimental techniques to determine Jones Matrix Jones phase microscopy (JPM) of transparent and anisotropic sample   Polarization Holographic Microscopy (PHM)

Jones phase microscopy (JPM) of transparent and anisotropic sample Basic Principle: The Jones matrix of sample is described as: Experimental setup: To retrieve complete Jones matrix +45 and -45 linearly polarized input beam is given to sample. Ref.- Zhuo Wang, Larry J. Millet, Martha U. Gillette, and Gabriel Popescu, Opt.Lett.33, (2008), 1270 Figure - Experimental setup of Jones Phase Microscopy [3], P0, PR and PA polarizer, C1, C2 collimating lenses, L1, L2 Fourier lens pair, BS beam splitter, CCD charge coupled device Requires Four Steps Not applicable for dynamic object

Polarization Holographic Microscopy (PHM) for Jones Matrix Imaging Figure - Experimental setup for Polarization Holographic Microscopy Ref.-Youngchan Kim, Joonwoo Jeong, Jaeduck Jang, Mahn Won Kim, and YongKeun Park, Opt.Exp.20, (2012), 9949.

Our Proposed Technique Advantages: Requires two steps In our proposed experimental technique, triangular Sagnac interferometer provides freedom to adjust carrier frequencies according to sampling law. Basic Principle: E’x =Jxx Ex +Jxy Ey E’y =Jyx Ex +Jyy Ey For +45 input beam

Our Proposed Technique (Double Shot) Experimental Setup: Basic Principle: Similarly for +45 input beam SF CCD - GX 2750 with specifications A/D 14-bit, 2750 × 2200 pixels and pixel pitch 4.54 μm. Figure - Experimental Setup of Jones matrix imaging; BS - Beam splitter, PBS - polarization beam splitter, M1, M2, M3 mirrors, HWP - half wave plate, SF spatial filter, CCD charge couple device, L1, L2, L3, L4,L5 lenses Ref.: N. K. Soni,  A. S. Somkuwar and R. K. Singh," Jones matrix imaging for transparent and anisotropic sample", Proc. SPIE9654, International Conference on Optics and Photonics 2015, 965420 (June 15, 2015); doi:10.1117/12.2181657; http://dx.doi.org/10.1117/12.2181657.  

Experimental Results: 1) Jones matrix for complete transmission (a) (b) (c) (d) Figure - Interferogram recorded for (a) +45 linearly polarized beam without sample (b)+45 linearly polarized beam with sample (c) -45 linearly polarized beam without sample (d) )-45 linearly polarized beam with sample

Fourier Fringe Analysis Technique Ex Ey F.T. I.F.T. Amplitude Two orthogonal polarization components are simultaneously retrieved through Fourier Fringe Analysis Phase

1) Jones matrix for complete transmission (b) (f) (c) (g) (d) (h) Figure - Jones matrix components for complete transmission ; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

2) Jones matrix for horizontal polarizer (b) Figure - Experimentally recorded hologram for (a) +45 (b)-45 illumination beam (a) (b) (c) (d) (e) (f) (g) (h) Figure - Jones matrix components for horizontal polarizer ; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

3) Jones matrix for vertical polarizer (b) Figure - Experimentally recorded hologram for (a) +45 (b)-45 illumination beam (a) (b) (c) (d) (e) (f) (g) (h) Figure - Jones matrix components for vertical polarizer ; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

4) Jones Matrix for quarter wave plate with fast axis aligned at horizontal (b) (a) Figure - Experimentally recorded interferogram for (a) +45 (b)-45 illumination beam (a) (b) (c) (d) (e) (f) (h) (g) Figure - Jones matrix components for QWP with fast axis aligned at horizontal ; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

Single Shot Jones Microscopy Limitations Of double shot Jones matrix imaging: Not applicable for dynamic objects Highly Sensitive for vibration and atmospheric air fluctuations Advantages of Single Shot Jones Microscopy: Requires only one measurement Applicable for dynamic objects also Capable to tune spatial frequencies No need of special optical elements like grating Experimental Setup: M1 M2 Y11 Y21 Y12 Y22 M4 Fourier spectrum M3 Figure-Single Shot Jones Microscopy

Experimental Results 1.Vertical polarizer Interferogram Amplitude of Fourier spectrum

1.Vertical polarizer (a) (b) (c) (d) (g) (e) (f) (h) 198 μm (a) (b) (c) (d) (g) (e) (f) (h) Figure - Jones matrix components for vertical polarizer ; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

2.Horizontal Polarizer Interferogram Amplitude of Fourier spectrum

2.Horizontal Polarizer (a) (b) (c) (d) (e) (f) (g) (h) 198 μm (a) (b) (c) (d) (e) (f) (g) (h) Figure - Jones matrix components for horizontal polarizer ; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

3.Polarizer @45 Interferogram Amplitude of Fourier spectrum

3.Polarizer @45 (a) (b) (c) (d) (e) (h) (f) (g) 396 μm (a) (b) (c) (d) (e) (h) (f) (g) Figure - Jones matrix components for polarizer at 45; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

4.Polarizer @-45 Amplitude of Fourier spectrum Interferogram

4.Polarizer @-45 (c) (a) (b) (d) (e) (f) (g) (h) 39.6 μm (c) (a) (b) (d) (e) (f) (g) (h) Figure - Jones matrix components for polarizer at -45; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

5.Half wave plate @0 Interferogram Amplitude of Fourier spectrum

5.Half wave plate @0 (a) (b) (c) (d) (e) (f) (g) (h) 198 μm (a) (b) (c) (d) (e) (f) (g) (h) Figure - Jones matrix components for half wave plate at 0 degree; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

6.Half wave plate @90 Amplitude of Fourier spectrum Interferogram

6.Half wave plate @90 (a) (b) (c) (d) (e) (f) (g) (h) 39.6 μm (a) (b) (c) (d) (e) (f) (g) (h) Figure - Jones matrix components for half wave plate at 90; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

7.Half wave plate @45 Amplitude of Fourier spectrum Interferogram

7.Half wave plate @45 (a) (b) (c) (d) (e) (g) (h) (f) 198 μm (a) (b) (c) (d) (e) (g) (h) (f) Figure - Jones matrix components for half wave plate at 45; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

8.QWP @0 Interferogram Amplitude of Fourier spectrum

8.QWP @0 (a) (d) (b) (c) (e) (f) (g) (h) 39.6 μm (a) (d) (b) (c) (e) (f) (g) (h) Figure - Jones matrix components for vertical polarizer ; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

9.QWP @90 Amplitude of Fourier spectrum Interferogram

9.QWP @90 (a) (b) (c) (d) (h) (e) (f) (g) 198 μm (a) (b) (c) (d) (h) (e) (f) (g) Figure - Jones matrix components for quarter wave plate at 90; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

10.PVA Amplitude of Fourier spectrum Interferogram

10. Polyvinyl Alcohol (PVA) (c) (d) (a) (b) (f) (g) (h) (e) 198 μm (c) (d) (a) (b) (f) (g) (h) (e) Figure - Jones matrix components for vertical polarizer ; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

11.Human Hair Interferogram Amplitude of Fourier spectrum

11.Human Hair (a) (c) (b) (d) (e) (f) (g) (h) Figure - Jones matrix components for human hair; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;

12.Blood Sample Amplitude of Fourier spectrum Interferogram

12. Jones matrix components for Blood Sample Figure - Jones matrix components for blood sample; (a), (c), (e), (g) amplitude distributions of Jxx, Jxy, Jyx, Jyy ; (b), (d), (f), (h) phase distributions of Jxx, Jxy, Jyx, Jyy ;