1. 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

2. 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)

3. 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

4. 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),

5. 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

6. 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,
(June 15, 2015); doi: / ; 

7. 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

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

9. 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 ;

10. 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 ;

11. 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 ;

12. 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 ;

13. 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

14. Experimental Results 1.Vertical polarizer Interferogram
Amplitude of Fourier spectrum

15. 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 ;

16. 2.Horizontal Polarizer
Interferogram
Amplitude of Fourier spectrum

17. 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 ;

18.
Interferogram
Amplitude of Fourier spectrum

19. 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 ;

20.
Amplitude of Fourier spectrum
Interferogram

21. 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 ;

22. 5.Half wave
Interferogram
Amplitude of Fourier spectrum

23. 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 ;

24. 6.Half wave
Amplitude of Fourier spectrum
Interferogram

25. 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 ;

26. 7.Half wave
Amplitude of Fourier spectrum
Interferogram

27. 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 ;

28.
Interferogram
Amplitude of Fourier spectrum

29. 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 ;

30.
Amplitude of Fourier spectrum
Interferogram

31. 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 ;

32. 10.PVA
Amplitude of Fourier spectrum
Interferogram

33. 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 ;

34. 11.Human Hair
Interferogram
Amplitude of Fourier spectrum

35. 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 ;

36. 12.Blood Sample
Amplitude of Fourier spectrum
Interferogram

37. 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 ;