1. Introduction

Anisotropic uniaxial materials are used widely for making optical devices such as wave plates, polarizers and beam-splitters [1,2]. A Rochon prism is a polarization beam-splitter that is formed by two cemented triangular wedges made from uniaxial birefringent materials. The optical axis of one of the wedges is parallel to the normal of incidence, and the optical axis of the other wedge is perpendicular to the plane of incidence, as is shown in Fig. 1. A linearly polarized incident beam is split into two orthogonal linearly polarized beams named the ordinary-extraordinary ray (oe ray) and the extraordinary-ordinary ray (eo ray) by the Rochon prism. The eo ray will not be deviated at normal incidence, while the oe ray is deviated by a splitting angle $\gamma$. The beam deflection angle is determined by the wedge angle of the Rochon prism and the refractive indices of the birefringent material. The propagation of the ordinary ray and the extraordinary ray in the uniaxial lossless prisms or slabs have been analyzed [3–9]. However, when absorption is taken into account, the theoretical description of light waves propagation in uniaxial media becomes rather complicated [10–20]. In this work we theoretically analyze inhomogeneous plane waves propagating in a Rochon prism made from uniaxial absorbing material. Expressions for the propagation directions of the ordinary and the extraordinary waves in the lossy Rochon prism are derived by complex ray-tracing method. As an application, the propagation characteristic of the inhomogeneous plane waves within the lossy Rochon prisms at normal incidence are analyzed and discussed.

 

Fig. 1 Ordinary and extraordinary rays in a lossless Rochon prism. The optical axes of the Rochon prism are such arranged that in the first wedge both of the ordinary and the extraordinary rays propagate along the optical axis at normal incidence, and in the second wedge the light rays propagate perpendicular to the optical axis. The ordinary ray (o ray) in the first wedge will be converted into the extraordinary ray (oe ray) in the second wedge, and the extraordinary ray (e ray) will be converted into the ordinary ray (eo ray), conversely.

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2. General properties of inhomogeneous waves in isotropic absorbing media

The isotropic absorbing medium is characterized by a complex index of refraction $\tilde{n}\text{=}n\text{-}i\kappa$, where $n$ is the real refractive index and $\kappa$ is the absorption coefficient. When a homogeneous plane wave coming from an isotropic lossless medium and incidents on the isotropic absorbing medium, the angle of incidence and the angle of refraction are related to each other by complex Snell-Descartes law [2,21–24]

The complex refraction angle ${\tilde{\theta }}_t$ determines a complex direction of light wave propagation in the absorbing medium and hence the refracted wave vector is also complex. The refracted wave vector $\tilde{K}$ can be written as

In general, the refracted waves are inhomogeneous and elliptically polarized plane waves. These elliptically polarized inhomogeneous plane waves can be described in the formalism of complex geometrical optics. With the complex notation for the field vectors (the electric field $\tilde{E}$, the displacement vector $\tilde{D}$, the magnetic field $\tilde{H}$, and the magnetic induction $\tilde{B}$), Maxwell’s equations adopt the form [16–20,25]

Unless otherwise specified, we will consider nonferromagnetic absorbing dielectric media without free charges ($\tilde{B}={\text{μ}}_{\text{0}}\tilde{H}$). The constitutive relation between the electric field $\tilde{E}$ and the displacement vector $\tilde{D}$ is given by

If we represent by $\tilde{F}$ any of the field vectors, then it can be written as [16–20,25]

Equation (8) represents any elliptically polarized inhomogeneous plane waves. The complex unitary vector ${\tilde{u}}_F$ completely determines the characteristics of the polarization of the field vector. The vector ${\tilde{u}}_F$ represents a superposition of two orthogonal elliptically polarized vibrations. The two orthogonal vibrations are determined by the real and imaginary parts of the vector ${\tilde{u}}_F$, respectively.

The inhomogeneity implies that the normal to the planes of constant phase and constant amplitude could be not coincide. The real direction of wave propagation is parallel to the normal to the plane of constant phase, and the direction of attenuation is parallel to the normal to the plane of constant amplitude. The directions of the wave propagation $k_{\text{Pr}}$and attenuation $k_{\text{At}}$, in the real space are given by [16–20,24]

The angle ${\alpha }_{\text{k}}$ between the wave propagation direction $k_{\text{Pr}}$ and the attenuation direction $k_{\text{At}}$ can be calculated from

3. Complex ray-tracing in Rochon prisms with absorption

3.1 Inhomogeneous plane-waves propagation in uniaxial anisotropic absorbing media

The dielectric permittivity $\tilde{\epsilon }$ is a complex tensor in an anisotropic absorbing medium. The complex dielectric permittivity tensor can be diagonalized into principal axes in the case of higher symmetry crystal classes. In the principal coordinate system (XYZ) where the Z-axis is parallel to the optical axis of the uniaxial absorbing medium, the dielectric permittivity tensor takes the form

The wave propagation unit vector $\tilde{k}$ obeys complex Fresnel’s equation, which in the principal coordinate system can be written as [16–20]

For a given complex direction of wave propagation $\tilde{k}$, there are two allowed orthogonally defined elliptically polarized inhomogeneous plane waves (the polarization is linear when the medium is lossless). One of the waves called an ordinary wave (o-wave), and another one called an extraordinary wave (e-wave). The complex refractive index of the o-wave is equal to the refractive index ${\tilde{n}}_o$, while the complex refractive index of the e-wave changes with the complex wave propagation direction ${\tilde{k}}_e$. The effective index of refraction ${\tilde{n}}_{eff}$ for the e-wave can be calculated from [20]

The electric displacement vector ${\tilde{D}}_o$ (or the unit vector ${\tilde{u}}_{{\text{D}}_o}$) and the electric field vector ${\tilde{E}}_o$ (or the unit vector ${\tilde{u}}_{{\text{E}}_o}$) of the o-wave are perpendicular to the complex plane determined by the unit wave vector ${\tilde{k}}_o$ and the optics axis $n_{\text{c}}$. However, the electric displacement vector ${\tilde{D}}_e$ (or the unit vector ${\tilde{u}}_{{\text{D}}_e}$) and the electric field vector ${\tilde{E}}_e$ (or the unit vector ${\tilde{u}}_{{\text{E}}_e}$) of the e-wave lie in the complex plane determined by the unit wave vector ${\tilde{k}}_e$ and the optics axis. Different from the ordinary wave, the unit e-wave vector ${\tilde{u}}_{E_e}$ is perpendicular to the complex ray propagation direction ${\tilde{s}}_e$, which could be not along the complex wave propagation direction, e.g., ${\tilde{u}}_{{\text{E}}_{\text{e}}}\cdot {\tilde{s}}_e\text{=}0$ and ${\tilde{u}}_{{\text{E}}_{\text{e}}}\cdot {\tilde{k}}_e\ne 0$.

The ray direction of the extraordinary wave is characterized by the complex Poynting vector${\tilde{S}}_e$, which is defined as [20]

The Poynting vector ${\tilde{S}}_e$ has a clear physical interpretation in the complex geometrical formalism. The propagation direction of the energy in the uniaxial absorbing medium is a function of time and the tip of the Poynting vector describes an ellipse determined by the unit Poynting vector and the effective index of refraction. The centre of the ellipses is given by the time-averaged value of the Poynting vector $〈{\tilde{S}}_e〉\text{=}\mathrm{Re}({\tilde{S}}_e)/2$.

3.2 Wave and ray directions of inhomogeneous plane-waves in lossy Rochon prism

A. Wave and ray directions of inhomogeneous plane-waves in the first wedge of the lossy Rochon prism

Considering a homogeneous plane wave coming from an isotropic lossless medium (Region I) and incidents on the entrance face of the lossy Rochon prism with an incident angle ${\theta }_i$, as is shown in Fig. 2(a). In Fig. 2(b), the ${\text{x}}_{\text{1}}{\text{y}}_{\text{1}}{\text{z}}_{\text{1}}$ coordinate system is chosen such that the ${\text{x}}_{\text{1}}{\text{z}}_{\text{1}}$ plane is the plane of incidence and both the normal to the interface $n_1$ and the optic axis $n_{c1}$ are in directions parallel to the ${\text{z}}_{\text{1}}$ axis. The complex direction cosines of the refracted ordinary wave in the first wedge (Region II) can be written as

 

Fig. 2 Ray tracing in a lossy Rochon prism. (a) Ordinary and extraordinary rays in the lossy Rochon prism. (b) Incident and refracted rays at the entrance face of the lossy Rochon prism. (c) Geometry of the interface between the two wedges in the lossy Rochon prism. (d) Incident and refracted rays at the exiting face of the lossy Rochon prism.

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The ray-tracing of the extraordinary wave in the first wedge (Region II) of the lossy Rochon prism is slightly complicated than that of the ordinary wave. The complex direction cosines of the extraordinary wave is given by

The effective index of refraction ${\tilde{n}}_{eff}$can be calculated from Eqs. (14) and (18) with $\tilde{\theta }\text{=}arccos({\tilde{k}}_e\cdot n_{\text{c1}})\text{=}{\tilde{\theta }}_e$, which gives a quadratic equation of $\cot {\tilde{\theta }}_e$, then [20]

As the optic axis $n_{\text{c1}}$ lies on the plane of incidence, the direction cosines of the extraordinary ray can be written as

B. Wave and ray directions of inhomogeneous plane-waves in the second wedge of the lossy Rochon prism

When the inhomogeneous plane-waves incident on the interface between the two lossy wedges in the Rochon prism, the ordinary wave (o wave) in the first wedge (Region II) will be refracted and converted into extraordinary wave (oe wave) in the second wedge (Region III), and the extraordinary wave (e wave) in the first wedge will be refracted and converted into ordinary wave (eo wave) in the second wedge, as is shown in Fig. 2(c). As the optic axis $n_{c2}$ of the second wedge is perpendicular to the plane of incidence, the refractive indices are ${\tilde{n}}_{eff}\equiv {\tilde{n}}_e$ and ${\tilde{n}}_o$ for the oe-wave and the eo-wave, respectively. In this special case, the wave and the ray directions of the oe-wave are coincide.

In the $x_2{y}_2{z}_3$ coordinate system where the $x_2{z}_2$plane is the plane of incidence and the normal to the interface$n_2$ is along the $z_2$ axis, the effective incident angles at the interface can be written as

Thus, the refraction angles for the oe-wave and the eo-wave in the second wedge are given by

Thus, the direction cosines ${\tilde{k}}_{oe}$ and ${\tilde{k}}_{eo}$ for the oe-wave and the eo-wave in the second wedge can be written as

C. Exiting angles of ordinary and extraordinary waves at the exiting face of the lossy Rochon prism

The ${\text{x}}_{\text{3}}{\text{z}}_{\text{3}}$ plane is the plane of incidence and the normal to the interface$n_3$ is along the $z_3$ axis, as is shown in Fig. 2(d). The effective incident angles at the exiting face of the second wedge are given by ${\tilde{\theta }}_{oe,eo}^i=\beta -{\tilde{\theta }}_{oe,eo}^{}$. Thus, the refracted angles (or the exiting angles) for the oe-wave and the eo-wave can be written as

4. Propagation characteristic of inhomogeneous plane waves within the lossy Rochon prisms at normal incidence

In this section, we will apply the expressions developed in the previous sections to analyze the propagation characteristic of the inhomogeneous plane waves within the lossy Rochon prisms at normal incidence. In this special case, the effective refraction index is ${\tilde{n}}_{eff}\equiv {\tilde{n}}_o$ for the e-wave in the first wedge of the prism. The o-wave and the e-wave propagate along the direction of $z_1$ and their field amplitudes are attenuated by the same term$A_t^{}=\exp [-\omega {\kappa }_o{\text{d}}_{\text{1}}/c]$ within the first wedge, where ${\text{d}}_{\text{1}}$ is the thickness of the first wedge.

The direction of the refracted e-wave (eo wave) in the second wedge is unchanged, and its field amplitude is attenuated by the term$A_t^{eo}=\exp [-\omega {\kappa }_o{\text{d}}_{\text{2}}/c]$ within the absorbing wedge, where ${\text{d}}_{\text{2}}$ is the thickness of the second wedge. The direction of the refracted o-wave (oe wave) will be deflected in the second wedge.

When the o-wave is incident on the interface between the two absorbing wedges, the refracted angle ${\tilde{\theta }}_{oe}$ in the second wedge is given by

The angle ${\tilde{\theta }}_{oe}$ is determined by the indices of refraction ${\tilde{n}}_{o,e}$ and the wedge angle $\beta$. The directions of the wave propagation $k_{\mathrm{Pr}}$ and attenuation $k_{\text{At}}$ can be calculated directly from Eqs. (9) and (10). In order to find the relations among the refractive indices and the wedge angle, the wave vector ${\widetilde{{\rm K}}}_{oe}$ can be written as the form

The real and the imaginary parts of the wave vector ${\widetilde{{\rm K}}}_{oe}$ obey

From Eqs. (28)-(31), we can get

From Eqs. (8), (29) and (33), we can get the attenuation term of the oe wave

In the Region IV, the transmitted oe wave vector ${\widetilde{{\rm K}}}_{oe}^t{}_{oe}$ can be written as

The real and the imaginary parts of the wave vector ${\widetilde{{\rm K}}}_{oe}^t{}_{oe}$ obey

From Eqs. (37)-(39), we can get

Equation (42) shows that the deflection angle (or splitting angle) ${\theta }_m^t$ of the Rochon prism is determined by the complex refractive indices of the uniaxial absorbing medium and the wedge angle.

We present numerical examples for the propagation characteristic of the inhomogeneous waves within the lossy Rochon prism at different wedge angles. The surrounding media are assumed to be air with $n_i=1$. Figures 3 and 6 show the apparent refractive indices ($n_m$ and ${\kappa }_m$) and the refracted angles (${\theta }_m$ and ${\psi }_m$) of the oe wave in the second wedge of the Rochon prism. The real refractive indices of the uniaxial material are$n_o\text{=}1.6$and $n_e\text{=}1.4$ but the absorption coefficients are different in Figs. 3 and 4. The real refractive indices of the uniaxial material are$n_e\text{=}1.4$ and $n_o\text{=}1.6$ but the absorption coefficients are different in Figs. 5 and 6. The attenuation directions are usually different from the wave propagation directions and the apparent absorption coefficients increase with the larger of wedge angles.

 

Fig. 3 Plots of the apparent refractive indices ($n_m$ and ${\kappa }_m$) and the refracted angles (${\theta }_m$ and ${\psi }_m$) for various values of wedge angle$\beta$ with the complex refractive indices of ${\tilde{n}}_o\text{=}1.6-0.5i$ and ${\tilde{n}}_e\text{=}1.4-0.5i$. Total reflection occurs at the angle of$\beta =\text{arc}\sin [\sqrt{n_e^{}{\kappa }_e^{}/n_o^{}{\kappa }_o^{}}\text{]=69}{\text{.3}}^{\circ }$.

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Fig. 4 Plots of the apparent refractive indices ($n_m$ and ${\kappa }_m$) and the refracted angles (${\theta }_m$ and ${\psi }_m$) for various values of wedge angle$\beta$ with the complex refractive indices of ${\tilde{n}}_o\text{=}1.6-1.0i$ and ${\tilde{n}}_e\text{=}1.4-1.0i$. Total reflection occurs at the angle of $\beta =\text{arc}\sin [\sqrt{n_e^{}{\kappa }_e^{}/n_o^{}{\kappa }_o^{}}\text{]=69}{\text{.3}}^{\circ }$.

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Fig. 5 Plots of the apparent refractive indices ($n_m$ and ${\kappa }_m$) and the refracted angles (${\theta }_m$ and ${\psi }_m$) for various values of wedge angle$\beta$ with the complex refractive indices of ${\tilde{n}}_o\text{=}1.4-0.5i$ and ${\tilde{n}}_e\text{=}1.6-0.5i$.

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Fig. 6 Plots of the apparent refractive indices ($n_m$ and ${\kappa }_m$) and the refracted angles (${\theta }_m$ and ${\psi }_m$) for various values of wedge angle$\beta$ with the complex refractive indices of ${\tilde{n}}_o\text{=}1.4-1.0i$ and ${\tilde{n}}_e\text{=}1.6-1.0i$.

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In Figs. 2 and 3, the refracted angles ${\theta }_m$ can be equal to $\text{π}/2$ at some special wedge angles. Considering the term $\text{4(}{n}_e^{}{\kappa }_e^{}-n_o^{}{\kappa }_o^{}{\sin }^2\beta )\text{=0}$ in the Eqs. (32) and (33), it requires ${\kappa }_e^{}={\kappa }_o^{}=0$ or ${\sin }^2\beta =n_e^{}{\kappa }_e^{}/n_o^{}{\kappa }_o^{}$(${\kappa }_o^{}\ne 0$, ${\kappa }_e^{}\ne 0$).

In the lossless case (${\kappa }_e^{}={\kappa }_o^{}=0$), Eqs. (32) and (33) can be reduce to

Equations (44)-(47) indicate that the refracted wave is the so-called evanescent wave that can propagate along the interface of two lossless media when $n_o^2{\sin }^2\beta \ge n_e^2$, which is plotted in Fig. 7.

 

Fig. 7 Plots of the apparent refractive indices ($n_m$ and ${\kappa }_m$) and the refracted angles (${\theta }_m$ and ${\psi }_m$) for various values of wedge angle$\beta$ with the refractive indices of ${\tilde{n}}_o\text{=}1.6-0.0i$ and ${\tilde{n}}_e\text{=}1.4-0.0i$. Total reflection occurs at the angle of $\beta =\text{arc}\sin [\sqrt{n_e^{}/n_o^{}}\text{]=69}{\text{.3}}^{\circ }$.

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In the absorbing case of ${\sin }^2\beta =n_e^{}{\kappa }_e^{}/n_o^{}{\kappa }_o^{}$, Eqs. (32) and (33) can be rewritten as

We can define an angle of total reflection ${\theta }_m\equiv \text{π}/2$ at the interface between the two absorbing media. When total reflection occurs, the apparent refractive indices and the refracted angles are given by

Equation (53) can be regard as the criteria for the total internal reflection of the e-wave at the interface between the two absorbing wedges at normal incidence in the Rochon prism.

Similarly, considering the term $\text{4(}{n}_e^{}{\kappa }_e^{}-n_o^{}{\kappa }_o^{}{\sin }^2\beta )$ is zero in the Eqs. (40) and (41), the total reflection can occurs at the exiting face of the Rochon prism when $n_i^2\le n_e^2{\sin }^2(\beta -{\theta }_m)$. And the apparent refractive indices and the refracted angles are given by

5. Conclusions

We theoretically analyze inhomogeneous plane waves propagating in the Rochon prisms made from uniaxial absorbing materials. Detailed expressions for the complex and real propagation and attenuation directions of the refracted inhomogeneous waves in the Rochon prisms are derived by complex ray-tracing method. As an application, the propagation characteristic of the inhomogeneous plane waves within the lossy Rochon prisms at normal incidence are analyzed and discussed.

The real wave propagation direction of the refracted inhomogeneous wave can be along the interface (${\theta }_m\equiv \text{π}/2$) between the two absorbing wedges of the Rochon prism. It can be regard as the total internal reflection for the absorbing media although the attenuation direction of the inhomogeneous plane is not perpendicular to the direction of wave propagation.

This work has concentrated on inhomogeneous plane waves propagation in the lossy Rochon prisms, however, further work will deal with the transmission and the polarization characteristics of the inhomogeneous plane waves in the lossy Rochon prisms.