## Analysis of the refraction of the extraordinary ray in a plane-parallel uniaxial plate with an arbitrary orientation of the optical axis.

Optics Express, Vol. 13, Issue 7, pp. 2549-2555 (2005)

http://dx.doi.org/10.1364/OPEX.13.002549

Acrobat PDF (180 KB)

### Abstract

Ray tracing formulas in a plane-parallel uniaxial plate bounded by an isotropic medium are analyzed when the crystal axis lies in the incident plane, and when its orientation is arbitrary. We present the behavior of the critical angle for the extraordinary ray as a function of the crystal axis position with respect to the normal to the refracting surface. We give the conditions in order to obtain the incidence angle at which the ordinary and extraordinary ray have the same refraction angle into the uniaxial crystal for particular positions of the optical crystal axis, also we give a condition for normal incidence in order to maximize or minimize the separation between the ordinary and extraordinary ray as a function of the optical crystal axis.

© 2005 Optical Society of America

## 1. Introduction

## 2. Ordinary and extraordinary ray tracing

**Z**axis as is shown in Fig. 1. Then

*ξ*

_{i}=0, and substituting into Eq. (1) from reference [8

8. M. Avendaño-Alejo, O. Stavroudis, and A. R. Boyain, “Huygens’ Principle and Rays in Uniaxial Anisotropic Media I. Crystal Axis Normal to Refracting Surface,” J. Opt. Soc. Am. A. **19**, 1668–1673, (2002). [CrossRef]

*η*

_{o}=sin

*θ*

_{o}and

*ζ*

_{o}=cos

*θ*

_{o}, or

*θ*

_{i}is the angle of incidence and

*θ*

_{o}is the refraction angle for the ordinary ray in the first interface[11].

**S**

_{e}=(

*ξ*

_{e}

*,η*

_{e}

*,ζ*

_{e}) depends on the direction cosines of the ordinary ray

*=(*

**S**_{o}*ξ*

_{o}

*,η*

_{o}

*,ζ*

_{o}), the direction cosines of the crystal axis

**A**=(

*α,β,γ*), and the ordinary and extraordinary refractive indices

*n*

_{o}

*, n*

_{e}respectively. In the particular case of a plane-parallel uniaxial plate, the first normal to the surface

**Z**

_{1}is parallel to the second normal to the surface

**Z**

_{2}; If we suppose that the crystal axis lies in the incidence plane then the direction cosines for the crystal axes take the values;

*α*=0,

*β*=sinϕ, and

*γ*=cos

*ϕ*, where

*ϕ*is the angle between the normal to the refracting surface and the crystal axis as shown in Fig. 1. Substituting these values in Eqs. (47–48) from Ref. [9

9. M. Avendaño-Alejo and O. Stavroudis, “Huygens’ Principle and Rays in Uniaxial Anisotropic Media II. Crystal Axis with Arbitrary Orientation,” J. Opt. Soc. Am. A. **19**, 1674–1679, (2002). [CrossRef]

## 3. Conditions for maximum and minimum separation between ordinary and extraordinary rays for normal incidence.

4. E. Cojocaru, “Direction cosines and vectorial relations for extraordinary-wave propagation in uniaxial media,” Appl. Opt. **36**, 302–306, (1997). [CrossRef] [PubMed]

5. Q.-T. Liang, “Simple ray tracing formulas for uniaxial optical crystal,” Appl. Opt. **29**, 1008–1010 (1990). [CrossRef] [PubMed]

*ϕ*or from Eq. (6) in order to obtain the critical points for the extraordinary ray direction as a function of the crystal’s axis position, thus we have;

*n*

_{o}=1.658,

*n*

_{e}=1.486, which correspond to calcite for λ=632[nm], then from Eq. (8) the angle for the crystal axis which maximizes the separation is approximately

*ϕ*≈42°. A plane-parallel uniaxial plate of calcite for normal incidence with this particular position of the crystal in Fig. 2 is shown. Substituting Eq. (8) into Eq. (6) we obtain the refraction angle for the extraordinary ray,

*n*

_{e}

*n*

_{o}), which give us the maximum separation between ordinary and extraordinary rays for normal incidence.

## 4. Applications

*θ*

_{o}and tan

*θ*

_{e}are given by Eqs. (2,9) respectively. These equations are a function of the refractive indices, angle of incidence and the crystal’s axis positions. The algebraic solution is too complicated. A numerical solution for the angle of incidence and the crystal axis position is shown in Fig. 3. Where for instance, we know the refractive indices values and for a given crystal axis orientation

*ϕ*Eq. (10) is solved for

*θ*

_{i}. An intent to obtain the condition from Eq. (10) is given by [14

14. A. A. Muryĭand and V. I. Stroganov, “Conditions for bringing the ordinary and extraordinary rays into coincidence in a plane-parallel plate fabricated from am optical uniaxial crystal,” J. Opt. Technol. **71**, 283–285, (2004). [CrossRef]

*n*

_{e}→

*n*

_{o}Eq. (9) reduces to Snell’s law Eq. (2).

## 5. Examples

*θ*

_{e}and

*θ*

_{o}for different positions of the crystal axis lying in the incidence plane. for

*n*

_{i}

*=n*

_{i2}=1, and the values for

*n*

_{o}and

*n*

_{e}given above.

### 5.1. Crystal axis at 30° to the normal of the refracting surface

*θ*

_{i}=55.9962°) the ordinary and extraordinary rays have the same refraction angle into the uniaxial crystal as was predicted by Eq. (10) and graphically is shown in Fig. 3. The critical angle for the extraordinary ray is

*θ*

_{eca}=38.9099° this is greater than the critical angle for the ordinary ray

*θ*

_{oca}=37.09°, see Table 1. The ray tracing process is shown in Fig. 4 for several cases for the angle of incidence.

### 5.2. Crystal axis at 60° to the normal of the refracting surface

## 6. Conclusions

## Acknowledgments

## References and links

1. | M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. |

2. | M. C. Simon, “Image formation through monoaxial plane-parallel plates,” Appl. Opt. |

3. | J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condes. Matter |

4. | E. Cojocaru, “Direction cosines and vectorial relations for extraordinary-wave propagation in uniaxial media,” Appl. Opt. |

5. | Q.-T. Liang, “Simple ray tracing formulas for uniaxial optical crystal,” Appl. Opt. |

6. | W.-Q. Zhang, “General ray-tracing formulas for crystal,” Appl. Opt. |

7. | G. Beyerle and I. S. McDermind, “Ray-tracing formulas for refraction and internal reflection in uniaxial crystal,” Appl. Opt. |

8. | M. Avendaño-Alejo, O. Stavroudis, and A. R. Boyain, “Huygens’ Principle and Rays in Uniaxial Anisotropic Media I. Crystal Axis Normal to Refracting Surface,” J. Opt. Soc. Am. A. |

9. | M. Avendaño-Alejo and O. Stavroudis, “Huygens’ Principle and Rays in Uniaxial Anisotropic Media II. Crystal Axis with Arbitrary Orientation,” J. Opt. Soc. Am. A. |

10. | M. Avendaño-Alejo and M. Rosete-Aguilar, “Paraxial Theory of birefringent lenses,” J. Opt. Soc. Am. A.22, No. 5, (2005) [CrossRef] |

11. | David Park, |

12. | J. P. Mathieu, |

13. | M. Avendaño-Alejo and M. Rosete-Aguilar, “Optical path difference in a plane parallel uniaxial plate,” unpublished. |

14. | A. A. Muryĭand and V. I. Stroganov, “Conditions for bringing the ordinary and extraordinary rays into coincidence in a plane-parallel plate fabricated from am optical uniaxial crystal,” J. Opt. Technol. |

**OCIS Codes**

(080.2710) Geometric optics : Inhomogeneous optical media

(160.1190) Materials : Anisotropic optical materials

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 2, 2005

Revised Manuscript: March 18, 2005

Published: April 4, 2005

**Citation**

M. Avendaño-Alejo, "Analysis of the refraction of the extraordinary ray in a plane-parallel uniaxial plate with an arbitrary orientation of the optical axis," Opt. Express **13**, 2549-2555 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2549

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### References

- M. C. Simon, �??Ray tracing formulas for monoaxial optical components,�?? Appl. Opt. 22, 354-360, (1983). [CrossRef] [PubMed]
- M. C. Simon, �??Image formation through monoaxial plane-parallel plates,�?? Appl. Opt. 27, 4176-4182, (1988). [CrossRef] [PubMed]
- J. Lekner, �??Reflection and refraction by uniaxial crystals,�?? J. Phys. Condes. Matter 3, 6122-6133, (1991). [CrossRef]
- E. Cojocaru, �??Direction cosines and vectorial relations for extraordinary-wave propagation in uniaxial media,�?? Appl. Opt. 36, 302�??306, (1997). [CrossRef] [PubMed]
- Q.-T. Liang, �??Simple ray tracing formulas for uniaxial optical crystal,�?? Appl. Opt. 29, 1008�??1010 (1990). [CrossRef] [PubMed]
- W.-Q. Zhang, �??General ray-tracing formulas for crystal,�?? Appl. Opt. 31, 7328�??7331, (1992). [CrossRef] [PubMed]
- G. Beyerle and I. S. McDermind, �??Ray-tracing formulas for refraction and internal reflection in uniaxial crystal,�?? Appl. Opt. 37, 7947�??7953 (1998). [CrossRef]
- M. Avendaño-Alejo, O. Stavroudis, A. R. Boyain, �??Huygens�?? Principle and Rays in Uniaxial Anisotropic Media I. Crystal Axis Normal to Refracting Surface,�?? J. Opt. Soc. Am. A. 19, 1668-1673, (2002). [CrossRef]
- M. Avendaño-Alejo, O. Stavroudis, �??Huygens�?? Principle and Rays in Uniaxial Anisotropic Media II. Crystal Axis with Arbitrary Orientation,�?? J. Opt. Soc. Am. A. 19, 1674-1679, (2002). [CrossRef]
- M. Avendaño-Alejo, M. Rosete-Aguilar, �??Paraxial Theory of birefringent lenses,�?? J. Opt. Soc. Am. A. 22, No. 5, (2005). [CrossRef]
- David Park, Classical Dynamics and its Quantum Analogues, second edition, (Springer Verlag, New York, 1990), pp. 18.
- J. P. Mathieu, Optics Parts 1 and 2, (Pergamon Press, New York, 1975), pp. 94.
- M. Avendaño-Alejo, M. Rosete-Aguilar, �??Optical path difference in a plane parallel uniaxial plate,�?? unpublished.
- A. A. Muryĭand and V. I. Stroganov, �??Conditions for bringing the ordinary and extraordinary rays into coincidence in a plane-parallel plate fabricated from am optical uniaxial crystal,�?? J. Opt. Technol. 71, 283-285, (2004). [CrossRef]

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