## Interferometer setup for the observation of polarization structure near the unfolding point of an optical vortex beam in a birefringent crystal |

Optics Express, Vol. 20, Issue 12, pp. 13573-13581 (2012)

http://dx.doi.org/10.1364/OE.20.013573

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### Abstract

We propose a novel birefringent interferometer setup for the study of unfolding points, and obtain for the first time to our knowledge the spatial polarization structure very near the unfolding point of a uniformly polarized optical vortex beam propagating in a birefringent crystal. The unfolding point is reconstructed by folding back the two separated eigen-beams at the output of the birefringent crystal into a single beam using another identical birefringent crystal, resulting in a birefringent interferometer of Mach-Zehnder type. We also demonstrate that the separation near the unfolding point can be varied by a small rotation of the second crystal.

© 2012 OSA

## 1. Introduction

10. C. Cheng, “The signal processing approach for the birefringent material based Mach-Zehnder interferometer design,” Proc. of IEEE, 48th Midwest Symposium on Circuits and Systems (Covington, Kentucky, 2005), 211–214, 10.1109/MWSCAS.2005.1594076. [CrossRef]

## 2. Theoretical model

*ê*and

_{H}*ê*are unit vectors in the horizontal and vertical directions respectively. The complex amplitudes

_{V}*E*and

_{e}*E*of the E-ray and O-ray respectively are given asin the coordinate system shown in Fig. 1 . Here

_{o}*w*is the beam radius and

_{0}*d*is the spatial separation between the two eigen-beams at the output of the BR-MZI. The small rotation of the second crystal will result in a spatial separation

*d*between the two eigen-beams due to the angle dependent refractive index of the extraordinary beam [12] inside the crystal. The beam propagation is considered in z direction.

_{0}, S

_{1}, S

_{2}and S

_{3}defined as [13]

_{0}are defined as

*s*= S

_{1}_{1}/S

_{0},

*s*= S

_{2}_{2}/S

_{0}and

*s*= S

_{3}_{3}/S

_{0}. Using the normalized Stokes parameters we construct the abstract fields

*s*and

_{1}+ is_{2}*s*. The argument of the abstract field

_{2}+ is_{3}*s*gives the information on the orientation of the major axis of polarization ellipse and the argument of

_{1}+ is_{2}*s*gives the phase difference between the two eigen-beams.

_{2}+ is_{3}## 3. Experimental details and results

_{1}). The diagonally polarized laser beam then enters a transmission type fork structured phase hologram (HG) made by electron beam lithography (EBL) [14]. The Laguerre-Gaussian (LG) beam with a topological charge of + 1 generated in the first order diffraction from the hologram is selected using an aperture (A). The size of the beam (2

*w*) at the detector plane is estimated to be 1.37 mm. The singly charged and diagonally polarized LG beam is then allowed pass through a birefringent crystal (Cr

_{0}_{1}) made of yttrium vanadate (YVO

_{4}) with dimensions 30 mm (length) X 12 mm (width) X 10 mm (height) (from CASTECH, China) and its optic axis in the horizontal plane making an angle of 45 degrees to the propagation direction.

*n*and

_{o}*n*of the crystals are given by 1.9929 and 2.2154 respectively. The birefringence (Δ

_{e}*n*) of the crystals and the walk-off angle (

*ρ*) between the two eigen-beams are calculated to be 0.2225 and 6.04 degrees respectively at 630 nm [15]. The two eigen-beams with orthogonal polarizations separated at the output of the first crystal due to birefringence are passed through a half wave plate (HWP) to interchange the polarizations of the two beams. The two beams after the HWP plate then enter a second crystal (Cr

_{2}) identical to the first one with its optic axis parallel to the first crystal and mounted on a rotational stage. The two eigen-beams are folded back together to form a single beam at the output of Cr

_{2}. Such an arrangement will result in a stable interferometer of Mach-Zehnder type.

_{2}) as shown in Fig. 1. The intensity distribution for different combinations of QWP and P

_{2}are recorded using a CCD (Model No. C5948, Hamamatsu) connected to a personal computer. As the BR-MZI is mounted on a translation stage it can be moved in and out of the input beam. Initially, the BR-MZI is moved out from the setup and we characterized the input beam. The polarization of the input OV beam is characterized by measuring the Stokes parameters using [17]where

*I*(

*β,γ*) is the output intensity when the axes of the QWP and P

_{2}are at

*β*and

*γ*respectively as measured from horizontal direction.

_{0}is normalized to its maximum value and S

_{1}, S

_{2}, S

_{3}are normalized to S

_{0}such that

*s*= S

_{1}_{1}/S

_{0},

*s*= S

_{2}_{2}/S

_{0}and

*s*= S

_{3}_{3}/S

_{0}. The CCD images shown in the present paper are inverted in the vertical direction as indicated by the arrow showing the + x direction in the figures. It can be seen from Fig. 2(a), 2(b), 2(c) and 2(d) that the polarization of the input beam is 45 degrees and the contribution of the horizontal/vertical and circular polarizations are minimal. The two angles 2α = arg (

*s*) and δ = arg (

_{1}+ is_{2}*s*) calculated from the Stokes parameters are shown in Fig. 2(e) and 2(f) respectively. The angle α is the azimuthal angle of the major axis of the polarization ellipse and δ is the phase difference between the horizontal and vertical components. Figure 2(e) and 2(f) confirm that the major axis of the polarization ellipse is at 45 degrees and there is no phase difference between the two orthogonal components.

_{2}+ is_{3}_{2}is rotated about its vertical axis (x-axis) to a maximum angle on one side where we see the two eigen-beams separated well in horizontal direction (y-axis) and then it is rotated to the other side in steps of one degree. At each rotation angle the output beam is captured through the CCD and we estimated the position of the O-ray and E-ray separately by calculating the center of gravity of the intensity distribution of each beam. The positions of the O-ray and E-ray at each rotation angle are plotted in Fig. 3(a) .

*d*of the two eigen-beams with respect to the input beam size is defined asHere

_{r}*d*is the separation between the two eigen-beams and 2

*w*is 1/e

_{0}^{2}beam width of the input beam at the detector plane. Figure 3(b) shows the separation

*d*of the two eigen-beams as a function of rotation angle of the Cr

_{2}. As the length and birefringence of the crystal (YVO

_{4}) used in our experiment is large compared to that of reference [7], we get relative separation of about 220% between the two eigen-beams at the output of a single crystal. This larger separation per crystal and larger angles of rotation of the crystal Cr

_{2}compared to reference [7] allows us to control the separation over a wide range of values.

_{2}. The zero value of the separation indicates the unfolding point and it is observed from Fig. 3(b) that this occurs between 346 and 347 degrees as read from meter of the rotational stage. The separation

*d*between the O-ray and E-ray is found to be around 30.2 μm and −29 μm at 346 degrees and 347 degrees respectively. The relative separation,

*d*corresponding to 346 degrees and 347 degrees is calculated using Eq. (5) to be 2.2% and −2.1% respectively at the output of the second crystal. The negative sign represents the interchange of the positions of the two-eigen beams after crossing the unfolding point. The dotted line in Fig. 3(b) represents the linear fit of the data. From the linear fit it is found that the separation changes −32.28

_{r}_{2}. We note that the exact angle for the unfolding point depends on the initial separation at the input of the second crystal, which in turn depends on the orientation of the first crystal.

_{2}is adjusted to the angle near the crossing point where the relative separation is measured to be 1.7% in the y-direction, which is very close to the unfolding point. The polarization of the output beam is characterized with the same method as for the input beam. The normalized Stokes parameters measured at the output of the BR-MZI are shown in Fig. 4 . It can be seen from Fig. 4(c) that in the upper half of the beam as displayed the polarization is 45 degrees, which is identical to the input beam polarization. However, the lower half of the beam is closer to right circular polarization (Fig. 4(d)). This variation can also be seen in Fig. 4(f) as a slope in δ, indicating relative tilt in the wave front of the two eigen-beams. Figure 4(e) shows that 2α is more or less constant, indicating that the major axis of the polarization ellipse remains near 45 degrees throughout the beam. Figure 4(b) shows that

*s*

_{1}is relatively small and varies in the direction from top left to bottom right. Since

*s*

_{1}is the balance between the intensities of the two eigen-beams, this variation is expected to coincide with the direction of the separation between the two eigen-beams, inferring that the two eigen-beams are separated in the x (vertical) direction as well as the y (horizontal) direction.

*d*and

_{1}*d*represent the separation between the two eigen-beams in the x and y directions respectively.

_{2}*θ*and

*ϕ*represent the spherical coordinates corresponding to the tilt angle and Δ is the overall phase difference between the two-eigen beams. Using Eq. (3) and Eq. (6) the normalized Stokes parameters are calculated and matched with experimental results. From the calculations,

*d*and

_{1}*d*are found to give a relative separation of 2.5% and 1.7% respectively, and

_{2}*θ*and

*ϕ*are estimated to be 35x10

^{−6}π and 0.1π respectively. The overall phase difference Δ between the two eigen-beams is found to be equal to 0.0012π. We note that the tilt can be a combined effect of a curved wave front and the separation between the two eigen-beams. The calculated results using Eq. (3) and Eq. (6) are shown in Fig. 5 . It can be observed that the calculated results coincide with the experimental results.

_{2}which results in a large separation between the two eigen-beams is also investigated. The crystal Cr

_{2}is rotated such that the rotation results in the relative separation of −42% in the y-direction between the eigen-beams at the output of the interferometer. The spatial distributions of the measured normalized Stokes parameters are shown in Fig. 6 . It can be observed that the large separation of the eigen-beams results in a non-uniform polarization across the beam cross section.

18. U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica **2**, 006 (2009), [CrossRef]

_{3}= 0, indicating linear polarization (L-line). In Fig. 6(e) two singular points can be observed which correspond to the positions where the major axis of the polarization ellipse is undefined. Such a situation will occur only for circular polarizations (C-points). The rotations of the major axis around the two C-points in Fig. 6(e) have opposite handedness.

*s*is slightly tilted from the horizontal direction, showing that the effect of separation in the y-direction is dominant but there is also a relatively small contribution from the separation in the x-direction. The separation in the x-direction is similar in magnitude to the one observed in Fig. 4 and Fig. 5, but opposite in sign. These vertical separations may be caused by small imperfections and misalignment in the experimental setup.

_{1}## 4. Conclusions

## Acknowledgments

## References and links

1. | M. S. Soskin and M. Vasnetsov, “Singular optics,” Prog. Opt. |

2. | J. F. Nye, |

3. | M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. |

4. | A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun. |

5. | M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., “Polarization singularities in the clear sky,” New J. Phys. |

6. | A. Desyatnikov, T. A. Fadeyeva, V. G. Shvedov, Y. V. Izdebskaya, A. V. Volyar, E. Brasselet, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express |

7. | F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. |

8. | F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express |

9. | X. D. Xu, P. K. Kuo, S. Y. Zhang, X. J. Shui, and Z. N. Zhang, “Application of an optical birefringence interferometer to photothermal detection,” Microw. Opt. Technol. Lett. |

10. | C. Cheng, “The signal processing approach for the birefringent material based Mach-Zehnder interferometer design,” Proc. of IEEE, 48th Midwest Symposium on Circuits and Systems (Covington, Kentucky, 2005), 211–214, 10.1109/MWSCAS.2005.1594076. [CrossRef] |

11. | J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature |

12. | A. Yariv and P. Yeh, |

13. | E. Collett, |

14. | Y. Miyamoto, M. Masuda, A. Wada, and M. Takeda, “Electron-beam lithography fabrication of phase holograms to generate Laguerre-Gaussian beams,” Proc. SPIE |

15. | http://www.castech.com/products_detail/&productId=61213567-1e08-41a0-9ac2-0d61b8c01db1.html. |

16. | T. Fadeyeva, Y. Egorov, A. Rubass, G. A. Swartzlander, Jr., and A. Volyar, “Indistinguishability limit for off-axis vortex beams in uniaxial crystals,” Opt. Lett. |

17. | T. Kihara, “Measurement method of Stokes parameters using a quarter-wave plate with phase difference errors,” Appl. Opt. |

18. | U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(230.5440) Optical devices : Polarization-selective devices

(260.1180) Physical optics : Crystal optics

(260.1440) Physical optics : Birefringence

(260.5430) Physical optics : Polarization

(050.4865) Diffraction and gratings : Optical vortices

(260.6042) Physical optics : Singular optics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: April 2, 2012

Revised Manuscript: May 19, 2012

Manuscript Accepted: May 24, 2012

Published: June 1, 2012

**Citation**

Maruthi M. Brundavanam, Yoko Miyamoto, Rakesh Kumar Singh, Dinesh N. Naik, Mitsuo Takeda, and Ken’ichi Nakagawa, "Interferometer setup for the observation of polarization structure near the unfolding point of an optical vortex beam in a birefringent crystal," Opt. Express **20**, 13573-13581 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13573

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### References

- M. S. Soskin and M. Vasnetsov, “Singular optics,” Prog. Opt.42, 219–276 (2001).
- J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (IoP Publishing, 1999).
- M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun.213(4-6), 201–221 (2002).
- A. Nesci, R. Dändliker, M. Salt, and H. P. Herzig, “Measuring amplitude and phase distribution of fields generated by gratings with sub-wavelength resolution,” Opt. Commun.205(4-6), 229–238 (2002).
- M. V. Berry, M. R. Dennis, and R. L. Lee, Jr., “Polarization singularities in the clear sky,” New J. Phys.6, 162 (2004).
- A. Desyatnikov, T. A. Fadeyeva, V. G. Shvedov, Y. V. Izdebskaya, A. V. Volyar, E. Brasselet, D. N. Neshev, W. Krolikowski, and Y. S. Kivshar, “Spatially engineered polarization states and optical vortices in uniaxial crystals,” Opt. Express18(10), 10848–10863 (2010).
- F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett.95(25), 253901 (2005).
- F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express14(23), 11402–11411 (2006).
- X. D. Xu, P. K. Kuo, S. Y. Zhang, X. J. Shui, and Z. N. Zhang, “Application of an optical birefringence interferometer to photothermal detection,” Microw. Opt. Technol. Lett.35(2), 140–143 (2002).
- C. Cheng, “The signal processing approach for the birefringent material based Mach-Zehnder interferometer design,” Proc. of IEEE, 48th Midwest Symposium on Circuits and Systems (Covington, Kentucky, 2005), 211–214, 10.1109/MWSCAS.2005.1594076. [CrossRef]
- J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature426(6964), 264–267 (2003).
- A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, 1984).
- E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).
- Y. Miyamoto, M. Masuda, A. Wada, and M. Takeda, “Electron-beam lithography fabrication of phase holograms to generate Laguerre-Gaussian beams,” Proc. SPIE3740, 232–235 (1999).
- http://www.castech.com/products_detail/&productId=61213567-1e08-41a0-9ac2-0d61b8c01db1.html.
- T. Fadeyeva, Y. Egorov, A. Rubass, G. A. Swartzlander, Jr., and A. Volyar, “Indistinguishability limit for off-axis vortex beams in uniaxial crystals,” Opt. Lett.32(21), 3116–3118 (2007).
- T. Kihara, “Measurement method of Stokes parameters using a quarter-wave plate with phase difference errors,” Appl. Opt.50(17), 2582–2587 (2011).
- U. T. Schwarz, F. Flossmann, and M. R. Dennis, “Topology of generic polarization singularities in birefringent crystals,” Topologica2, 006 (2009), [CrossRef]

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