## A unit structure Rochon prism based on the extraordinary refraction of uniaxial birefringent crystals |

Optics Express, Vol. 21, Issue 11, pp. 13162-13168 (2013)

http://dx.doi.org/10.1364/OE.21.013162

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### Abstract

Based on the Fermat's principle, the universal theory of refraction and reflection of extraordinary rays (e-rays) in the uniaxial crystal is formulated. Using this theory, a new unit structure prism is designed, and its properties are studied. Based on the theoretical results, such a prism is achieved experimentally by using the Iceland crystal. In both theoretical and experimental studies, this new prism shows excellent polarization splitting performances such as big and adjustable splitting angle, comparing to the conventional Rochon prism. For the sample prism with the optical axis angle of 45°, the splitting angle reaches 19.8°in the normal incidence, and the maximum splitting angle reaches 28.44° while the incidence angle is −4°.

© 2013 OSA

## 1. Introduction

1. S. Gräf, G. Staupendahl, C. Seiser, B.-J. Meyer, and F. A. Müller, “Generation of a dynamic polarized laser beam for applications in laser welding,” J. Appl. Phys. **107**(4), 043102 (2010). [CrossRef]

## 2. Universal formula of e-rays refraction and reflection in uniaxial birefringent crystals

_{o}and n

_{e}) is on the right side. The direction cosine of the optical axis is (cosα

_{0}, cosγ

_{0}), where cosα

_{0}= sinγ

_{0}. The incident medium (the refractive index is n) is isotropically on the left side. The incident light into the crystal is on point O from point A on the xOz plane and the incident angle is i. The refraction of e-rays light is on point B from point O on the xoz plane and the refraction angle is γ

_{R}. The incident light vector is

_{i}, cosγ

_{i}) is the direction cosine of the incident light, and

_{R}, cosγ

_{R}) is the direction cosine of light refraction, and

_{R}is the angle between the e-rays and the optical axis. By comparing this equation with Fermat’s principle in the isotopic crystal (

_{R}

^{’}within the uniaxial crystal can be obtained by

_{A}= l

_{Ax}cosα

_{i}+ l

_{Az}cosγ

_{i}, and δl

_{Az}= 0, the following equation can be obtained:similarlybased on cosα

_{R}= l

_{Bx}/l

_{B}, cosγ

_{R}= l

_{Bz}/l

_{B}, cosψ

_{R}= cosα

_{0}cosα

_{R}+ cosγ

_{0}cosγ

_{R}, the following is obtainedusing the variation about Eq. (3) and sinψ

_{R}= -δcosψ

_{R}, Eq. (7) is obtained.since A and B are two fixed points, δl

_{Bx}= -δl

_{Ax}andδl

_{Bz}= -δl

_{Az}should be developed, and if Eqs. (2), (4), (5) and (6) are placed in Eq. (7), the following can be obtained

_{i}= i, cosα

_{i}= sinγ

_{i}, cosα

_{0}= sinγ

_{0}, cosα

_{R}= sinγ

_{R}, Eq. (9) can be obtained from Eq. (8)which f(ψ

_{R}) = n

_{R}

^{’}/(n

_{e}cosψ

_{R}).

^{2}α

_{R}+ sin

^{2}γ

_{R}= 1, and cos

^{−2}ψ

_{R}= tan

^{2}ψ

_{R}+ 1 = f

^{2}(ψ

_{R}) + (1-n

_{o}

^{2}/n

_{e}

^{2}), a quadratic equation of f(ψ

_{R}) is obtained:

_{R}in Eq. (9) with sinγ

_{R}, divide respectively both sides of Eqs. (9) and (10), and use Eq. (12). The universal refraction formula angle of e-rays from the isotropic medium to the uniaxial crystal can be obtainedWhen the e-rays are incident from the uniaxial crystal to the isotropic medium, the formula of p-polarization refraction angle i

_{e}in an isotropic medium is expressed as

_{i}is the incidence angle of e-rays in uniaxial crystals and n

_{i(γ0)}

^{2}= n

_{R(γ0)}

^{2}= n

_{e}

^{2}cos

^{2}γ

_{0}+ n

_{o}

^{2}sin

^{2}γ

_{0}.

_{0}, cosγ

_{0}), where ψ

_{i}is the angle between the incident e-rays and the optical axis, (cosα

_{ri}, cosγ

_{ri}) is the direction cosine of the incident light, and (cosα

_{rr}, cosγ

_{rr}) is the direction cosine of the reflected light.

## 3. Prism design and light path analysis

_{0}= γ

_{0}= π/4. The angle value from the rays to the normal is positive for clockwise and negative for anticlockwise. The o-rays’ incidence angle from the air is set equal to the emergence angle from the prism, namely i

_{0}= i, with identical signs. The propagation of e-rays, such as the refraction angle of e-rays from the prism, must be analyzed.

_{1}, the refraction angle of e-rays is γ

_{R}. Since the light incident from air is n

_{R(γ0)}

^{2}= (n

_{e}

^{2}+ n

_{o}

^{2})/2, and n = 1, Eq. (3) can be transformed into

_{2,}the total reflection of e-rays is shown in Fig. 3, γ

_{ri}= |γ

_{R}|, α

_{ri}= π/2-γ

_{ri}= π/2-|γ

_{R}|, so tanα

_{ri}= tan(π/2-|γ

_{R}|) = |cotγ

_{R}|, substitutes the above results into Eq. (15)

_{3}, the γ

_{R}of Eq. (14) is the

_{i}= γ

_{rr}= π/2-α

_{rr}, so

_{e}) and the incidence angle (i) can be determined by Eqs. (16), (18), and (19). Given that the output angle of o-rays i

_{o}= i, the splitting angle of the new prism is

6. Z. Liu, Z. F. Lin, and S. T. Chui, “Negative refraction and omnidirectional total transmission at a planar interface associated with a uniaxial medium,” Phys. Rev. B **69**(11), 115402 (2004). [CrossRef]

9. K. Sinchuk, R. Dudley, J. D. Graham, M. Clare, M. Woldeyohannes, J. O. Schenk, R. P. Ingel, W. Yang, and M. A. Fiddy, “Tunable negative group index in metamaterial structures with large form birefringence,” Opt. Express **18**(2), 463–472 (2010). [CrossRef] [PubMed]

## 4. Experiments and results

## 5. Conclusion

## Acknowledgments

## References and links

1. | S. Gräf, G. Staupendahl, C. Seiser, B.-J. Meyer, and F. A. Müller, “Generation of a dynamic polarized laser beam for applications in laser welding,” J. Appl. Phys. |

2. | H. J. Cornelissen, H. P. M. Huck, D. J. Broer, S. J. Picken, C. W. M. Bastiaansen, E. Erdhuisen, and N. Maaskant, “38.3: Polarized light LCD backlight based on liquid crystalline polymer film: A new manufacturing process,” S/D Symposium Dig. Tech. Pap. |

3. | J. Fatome, S. Pitois, P. Morin, and G. Millot, “Observation of light-by-light polarization control and stabilization in optical fibre for telecommunication applications,” Opt. Express |

4. | J. N. Damask, |

5. | W. Wang, F. Q. Wu, and F. F. Su, “Symmetric polarization beamsplitting prism based on three-element Wollaston prism,” Opt. Technol. |

6. | Z. Liu, Z. F. Lin, and S. T. Chui, “Negative refraction and omnidirectional total transmission at a planar interface associated with a uniaxial medium,” Phys. Rev. B |

7. | Q. Cheng and T. C. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B |

8. | J. Sun, L. Liu, G. Dong, and J. Zhou, “Efficient polarization beam splitter based on an indefinite medium,” J. Electromagn. Waves Appl. |

9. | K. Sinchuk, R. Dudley, J. D. Graham, M. Clare, M. Woldeyohannes, J. O. Schenk, R. P. Ingel, W. Yang, and M. A. Fiddy, “Tunable negative group index in metamaterial structures with large form birefringence,” Opt. Express |

**OCIS Codes**

(230.1360) Optical devices : Beam splitters

(230.5440) Optical devices : Polarization-selective devices

**ToC Category:**

Optical Devices

**History**

Original Manuscript: February 28, 2013

Revised Manuscript: May 3, 2013

Manuscript Accepted: May 7, 2013

Published: May 22, 2013

**Citation**

Wendi Wu, Fuquan Wu, Meng Shi, Fufang Su, Peigao Han, and Lili Ma, "A unit structure Rochon prism based on the extraordinary refraction of uniaxial birefringent crystals," Opt. Express **21**, 13162-13168 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13162

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### References

- S. Gräf, G. Staupendahl, C. Seiser, B.-J. Meyer, and F. A. Müller, “Generation of a dynamic polarized laser beam for applications in laser welding,” J. Appl. Phys.107(4), 043102 (2010). [CrossRef]
- H. J. Cornelissen, H. P. M. Huck, D. J. Broer, S. J. Picken, C. W. M. Bastiaansen, E. Erdhuisen, and N. Maaskant, “38.3: Polarized light LCD backlight based on liquid crystalline polymer film: A new manufacturing process,” S/D Symposium Dig. Tech. Pap.35(1), 1178–1181 (2004). [CrossRef]
- J. Fatome, S. Pitois, P. Morin, and G. Millot, “Observation of light-by-light polarization control and stabilization in optical fibre for telecommunication applications,” Opt. Express18(15), 15311–15317 (2010). [CrossRef] [PubMed]
- J. N. Damask, Polarization Optics in Telecommunications (Springer, 2005).
- W. Wang, F. Q. Wu, and F. F. Su, “Symmetric polarization beamsplitting prism based on three-element Wollaston prism,” Opt. Technol.30, 182–186 (2004).
- Z. Liu, Z. F. Lin, and S. T. Chui, “Negative refraction and omnidirectional total transmission at a planar interface associated with a uniaxial medium,” Phys. Rev. B69(11), 115402 (2004). [CrossRef]
- Q. Cheng and T. C. Cui, “Negative refractions in uniaxially anisotropic chiral media,” Phys. Rev. B73(11), 113104 (2006). [CrossRef]
- J. Sun, L. Liu, G. Dong, and J. Zhou, “Efficient polarization beam splitter based on an indefinite medium,” J. Electromagn. Waves Appl.26(11-12), 1423–1431 (2012). [CrossRef]
- K. Sinchuk, R. Dudley, J. D. Graham, M. Clare, M. Woldeyohannes, J. O. Schenk, R. P. Ingel, W. Yang, and M. A. Fiddy, “Tunable negative group index in metamaterial structures with large form birefringence,” Opt. Express18(2), 463–472 (2010). [CrossRef] [PubMed]

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