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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 Optical Society of America
1. Introduction
Polarization optics has been the subject of intense research since the eighteenth century. The polarization characteristics of lasers have been applied to industrial manufacturing, communication, and laser measurement [1–4]. The linearly polarized lights are mainly produced by a polarization splitter, which is made out of uniaxial birefringent crystals in the form of prisms. Most of the current polarization splitting prism is made of multi-structures by Iceland crystals [5], which greatly impairs the mechanical characteristics and the power damage threshold of the prism. To overcome these limitations, a new unit structure prism has been designed on the basis of e-rays propagation property in uniaxial crystals.
2. Universal formula of e-rays refraction and reflection in uniaxial birefringent crystals
The e-rays refraction and reflection equations in uniaxial birefringent crystal are deduced to understand the e-rays light propagation behavior. The optical axis and incidence light in the xoz plane are shown in Fig. 1. The x axis is the surface boundary between the isotropic medium and the crystal. The uniaxial crystal (the principal refractive indexes are no and ne) is on the right side. The direction cosine of the optical axis is (cosα0, cosγ0), where cosα0 = sinγ0. 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 γR. The incident light vector is, where (cosαi, cosγi) is the direction cosine of the incident light, and . The e-rays light vector is in the uniaxial crystal, where (cosαR, cosγR) is the direction cosine of light refraction, and.
Fermat’s principle of e-rays within uniaxial crystals is obtained from the metric optics
where ψR is the angle between the e-rays and the optical axis. By comparing this equation with Fermat’s principle in the isotopic crystal (), the refractive index of e-rays nR’ within the uniaxial crystal can be obtained byBased on Fermat principle, the extreme value of e-rays passing through point A and point B can be found
by analyzing differential for lA = lAxcosαi + lAzcosγi, and δlAz = 0, the following equation can be obtained:similarlybased on cosαR = lBx/lB, cosγR = lBz/lB, cosψR = cosα0cosαR + cosγ0cosγR, the following is obtainedusing the variation about Eq. (3) and sinψR = -δcosψR, Eq. (7) is obtained.since A and B are two fixed points, δlBx = -δlAx andδlBz = -δlAz should be developed, and if Eqs. (2), (4), (5) and (6) are placed in Eq. (7), the following can be obtainedIt has been assumed that Axis x is on the interface between two mediums, Axis z is the normal line and crystal optical axis in Plane xOz. Therefore, the refracted e-rays should be in Plane xOz, and thus γi = i, cosαi = sinγi, cosα0 = sinγ0, cosαR = sinγR, Eq. (9) can be obtained from Eq. (8)
which f(ψR) = nR’/(necosψR).Equations (8) and (9) still contain an uncertain variable: ψR, the angle between crystal e-rays and optical axis, and it is interrelated with angle αR. Therefore, is Eq. (9) is placed in cosψR = cosα0cosαR + cosγ0cosγR, the following is developed:
where nR2(γ0) = ne2cos2γ0 + no2sin2γ0.Equation (9) squared plus Eq. (10) squared and based on the equations cos2αR + sin2γR = 1, and cos−2ψR = tan2ψR + 1 = f2(ψR) + (1-no2/ne2), a quadratic equation of f(ψR) is obtained:
The solution of the above equation is
Replace cosαR in Eq. (9) with sinγR, 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 ie in an isotropic medium is expressed aswhere γi is the incidence angle of e-rays in uniaxial crystals and ni(γ0)2 = nR(γ0)2 = ne2cos2γ0 + no2sin2γ0 .The origin of the coordinate moves to the light reflection point, in order to analyze the condition of e-rays reflection in uniaxial crystals, as shown in Fig. 2. The z axis places on the bounding surface between the uniaxial crystal (above z axis) and the isotropic medium (under z axis). The direction cosine of the optical axis is (cosα0, cosγ0), where ψi is the angle between the incident e-rays and the optical axis, (cosαri, cosγri) is the direction cosine of the incident light, and (cosαrr, cosγrr) is the direction cosine of the reflected light.
Similar to the analytic process of e-rays refraction, the reflection formula of the e-rays in uniaxial crystals is expressed as
where nr(α0)2 = ne2sin2α0 + no2cos2α0 .3. Prism design and light path analysis
Figure 3 illustrates the structure and the coordinate system of the new prism that is made by a uniaxial negative crystal. The double arrow is the optical axis of the uniaxial crystal. Surfaces 1, 2, and 3 are the optical polished surfaces. When monochromatic natural light is incident in the uniaxial crystal, it will separate into ordinary rays (o-rays) and e-rays. The o-rays pass through the crystal and exit on Surface 2 without any reflection. The e-rays generate a total reflection on Surface 3 and the reflection light exits the crystal on Surface 2 with a different angle from o-rays. Given that the light splitting effect is similar to the conventional Rochon prism, the new prism is called the unit structure Rochon prism.
The heavy lines represent the principal section. For simplicity, let α0 = γ0 = 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 i0 = i, with identical signs. The propagation of e-rays, such as the refraction angle of e-rays from the prism, must be analyzed.
On the incident surface of x1, the refraction angle of e-rays is γR. Since the light incident from air is nR(γ0)2 = (ne2 + no2)/2, and n = 1, Eq. (3) can be transformed into
On coordinate x2, the total reflection of e-rays is shown in Fig. 3, γri = |γR|, αri = π/2-γri = π/2-|γR|, so tanαri = tan(π/2-|γR|) = |cotγR|, substitutes the above results into Eq. (15)
On the output surface of x3, the γR of Eq. (14) is the, so Eq. (14) is changed into
where, “+” is the uniaxial negative crystal and“-” is the uniaxial positive crystal.Since γi = γrr = π/2-αrr, so
For light of certain wavelength through a unit structure Rochon prism, as shown in Fig. 3, the relationship between the refraction of e-rays (ie) and the incidence angle (i) can be determined by Eqs. (16), (18), and (19). Given that the output angle of o-rays io = i, the splitting angle of the new prism is
An Iceland crystal is a uniaxial negative crystal. It is one of the most widely used materials in polarization devices. In Eqs. (3), (7), and (10), the negative refraction occurs in the range of a positive angle [6–9], which is a natural disposition of uniaxial crystals, as shown in Fig. 3, and the new prism uses the property that negative refraction occurs when incident angle is positive. The prism is designed to work within a range of ± 4° to realize negative refraction and total reflection. When the incidence angle equals 0°, the relationship between the splitting angle and the wavelength of laser source is shown in Fig. 4.
Fig. 4 Variation curve of splitting angle relationship with the change of incident light wavelength.
The unit structure Rochon prism produces a large splitting angle of 19.788°, with an incidence angle of 0°, making it two or three times larger than the conventional Rochon prism. The splitting angle of the prism varies in a large range. When the incidence angle varies by 1°, the splitting angle is varies by 2°. The splitting angle φ tapers as the wavelength of incidence light increases. The variation of change tapers when the wavelength of light changes from short wave to long wave.
The light incidence angle is 0°. Figure 5 illustrates the changes of the splitting angle with the variation of optical axis angle. When the optical axis angle is 46.5°, the splitting angle reaches its maximum value of 19.8°.
4. Experiments and results
Good quality Iceland crystals are used as a primary material to produce the sample prism with an optical axis angle of 45°. In the test of the polarization degree of each emergent light from the new prism, the light source is a laser operating wavelength at 532 nm with an output power of 30 mW. A Glan-Thompson prism is used to analyze its polarization, and the result is more than 0.9999 degree of polarization. In the case of uncoating, the maximum loss of e-rays is 10%, and that of o-rays is 13%.
A splitting angle test system is determined by using a goniometer, with 633-nm and 532-nm laser sources. The splitting angles are tested with varying incidence angles. The results of the tests are shown in Table 1, which are in accordance with the theories.
Table 1. Test results using a 633nm and 532nm laser source.
5. Conclusion
The universal formula of refraction and reflection of e-rays in uniaxial crystals are derived from Fermat’s principle. A new unit structure splitting beam prism is designed on the basis of the theory. Given the crystal intrinsic properties, the splitting angle of the prism is larger in the short-wavelength than in the long-wavelength. By the test of the sample new prism, the degree of polarization of each emergent light is larger than 0.9999, make an optical axis angle of 45°, a splitting angle of 19.8° when the incidence angle is 0°, and a largest splitting angle of 28.44° when the incidence angle is −4°. The experimental results are in accordance with the theories.
Acknowledgments
This research is supported by the National Nature Science Foundation of China (No.11104161 and No. 11104162)
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