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Abstract
The paper proposes a Wollaston-type crystal polarizer suitable for broadband operation within the visible spectral band up to the far infrared band based on unique optical materials, mercurous halides (Hg2X2). This paper introduces the general characteristics and optical properties of these birefringent tetragonal optical materials, as well as the general description of a Wollaston prism and the process of its parameter optimization. In general, the Wollaston polarizer is constructed from two combined wedge-shaped prisms. The key parameters that affect the properties of the Wollaston polarizer are then the cut angle of these two prisms and the refractive index of the exploited optical cement (immersion) that bonds the prisms together. The optimal prism cut angles and immersion refractive index are investigated to maximize the Wollaston parameters, such as the transmittance of the polarized radiation and the separation angle of the output orthogonally polarized beams. This process is significantly dependent on the characteristics of all selected mercurous halides (Hg2Cl2, Hg2Br2, Hg2I2). The optimal values of the prism cut angle for each material are selected based on the outlined results. In addition, the Wollaston prism behaviour regarding real radiation propagation is modelled in detail via the Zemax optical studio. The presented models aim to aid in the real design and fabrication of a broadband Wollaston polarizer based on mercurous halides.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Today, many applications in optics and photonics exploit the polarization phenomenon of electromagnetic radiation [1–5]. The simplest way to obtain and operate with polarized radiation is using a standard linear or circular type of polarizer [6]. There are several types of these devices, such as wire-grid, thin-film, or birefringent crystal polarizers, each with its specific usage [7,8]. The thin-film and wire-grid polarizers are not commonly optimized for broadband operation and focus on specific spectral bands. Aiming mainly at the infrared spectral bands, the menu of usable polarizing devices thickens. Although wire grid polarizers are applicable in the TIR (thermal infrared) spectral band from $7\;\mathrm {\mu }\textrm {m}$ to $15\;\mathrm {\mu }\textrm {m}$, they usually do not offer a superior degree of polarization and are mainly specialized on narrower bands. Therefore, crystal polarizers are most commonly used to achieve a superior degree of polarization for radiation propagation through the polarizer across as wide spectral range as possible. Nevertheless, the usage of such a polarizer is limited by selecting a suitable optical material that can be used for a polarizer fabrication. Among the materials that are used most frequently for polarizer fabrication is standard calcite ($\mathrm {Ca}\mathrm {CO}_3$), with transparency ranging between $0.3 \;\mathrm {\mu }\textrm {m}$ and $2.3 \;\mathrm {\mu }\textrm {m}$ [9], which covers the VIS (visible), NIR (near-infrared), and SWIR (short-wave infrared) spectral bands. The other frequently commercially used material magnesium fluoride ($\mathrm {Mg}\mathrm {F}_2$) is transparent in an even wider spectral range of $0.2 \;\mathrm {\mu }\textrm {m}$ - $6 \;\mathrm {\mu }\textrm {m}$ and additionally covers a MWIR (mid-wave infrared) band [10]. However, there are no commercially used crystal polarizers that would cover VIS-TIR spectral bands and even parts of the FIR (far-infrared) band. Highly promising optical materials capable of filling this void are mercurous halides ($\mathrm {Hg}_2\mathrm {X}_2$), such as calomel ($\mathrm {Hg}_2\mathrm {Cl}_2$), kuzminite ($\mathrm {Hg}_2\mathrm {Br}_2$), and moschelite ($\mathrm {Hg}_2\mathrm {I}_2$) [11]. These materials offer exceptional transparency from $0.35 \;\mathrm {\mu }\textrm {m}$ to $40 \;\mathrm {\mu }\textrm {m}$ and birefringence up to 1.48.
Several proposals have been made for the use of mercurous halides as polarizers in the past [12,13], but most research focuses on their extraordinary acousto-optical properties and applications [14–22]. For example, [23] analyzed in detail the characteristic of mercurous halides regarding their acousto-optical properties, such as the acousto-optic figure of merit, but their polarizing behaviour important for the polarizer construction has been omitted. Nevertheless, the current problem of mercurous halides is with their complicated growth, polishing treatment, and quality evaluation [24–30]. Moreover, for the successful fabrication of a polarizer based on mercurous halides, a suitable design scheme for precise cutting and polishing of the growth mercurous halide crystal has to be introduced.
In the past years, there has been a short introduction regarding the exploitation of mercurous halide crystalline materials as polarizers [12]. However, detailed analysis and optimization schemes on how the polarizers should be designed have been omitted. Recently, a study on Glan-type polarizer based on mercurous halides was proposed. [13]. The authors investigated their own design of a Glan-type polarizer, but mainly for kuzminite and primarily with a focus on the acceptance angles. Reviewing these studies up to the present, a detailed design and construction study focusing on a standard Wollaston polarizer based on mercurous halides has been almost completely omitted.
Contrary to the Glan-type polarizers, the Wollaston polarizer usually needs for its operation a specified optical cement (adhesive immersion) that bonds together the two fundamental Wollaston crystal wedge prisms. Finding a suitable refractive index of immersion, together with a correct cut angle of the wedge prisms, makes analyzing and designing the Wollaston prism more difficult.
In general, Wollaston polarizers find applications in various optical fields, such as polarimetry [31–35], microscopy [36,37], and spectroscopy [38,39]. Taking advantage of a mercurous halide-based Wollaston prism and using it for the mentioned applications may enrich them and extend these fields to a broader spectral range. Specifically, this type of polarizer could be used efficiently for thermal hyperspectral imaging systems [23]. Nevertheless, for all these applications, a precise design of this type of Wollaston polarizer is necessary to optimize its performance in specified spectral bands.
Considering the current state of the mercurous halide-based polarizers described above, this paper presents an analysis and proposes design parameters for the mercurous halide-based Wollaston prism with the aim of maximizing the transmittance and the output beam separation angle over as wide spectral range as possible. For this analysis, a high focus is given to the maximal exploitation of the unique material properties of mercurous halides. Suitable cut angles of the Wollaston wedge prisms and a convenient refractive index of the necessary optical cement are proposed and selected. The analysis is performed separately for all mercurous halides, calomel, kuzminite, and moschelite. Following the cut angle and immersion selection, a ray-tracing of the mercurous halide Wollaston prism based on the Zemax optic studio is performed.
The paper is structured into five main sections. The introduction is followed by the general characterization of mercurous halides in Section 2. A detailed description of the optimized Wollaston parameters is presented in Section 3. Section 4 presents the analysis and ray-tracing results and provides a subsequent discussion on this topic. The paper is then summarized in conclusion Section 5.
2. Mercurous halide characteristics
Fundamentally, mercurous halides are ranked among tetragonal uniaxial anisotropic crystals. To evaluate the optical polarization behaviour of these materials, it is convenient to describe how they affect the electric field of electromagnetic radiation. In general, the electric field in any linear dielectric anisotropic material can be expressed [6] as
where $\mathbf {D}$ is a vector of electrical induction, $\mathbf {E}$ represents a vector of electrical intensity, and $\boldsymbol {\epsilon }^{-1}$ is a tensor of electrical impermeability. Thus, the dielectric properties of the material are fully described by the impermeability tensor. For an anisotropic material with a certain symmetry in a principal coordinate system $xyz$, Eq. (1) can be rewritten asThe undeniable advantage of mercurous halides is their high positive birefringence. At a wavelength $0.633 \; \mathrm {\mu }\textrm {m}$ the ordinary and extraordinary refractive indices of calomel reach values equal to $n_{\text {o}} =1.96,\; n_{\text {e}} = 2.61$. Furthermore, the refractive indices of kuzminite and moschelite reach values $n_{\text {o}} =2.12,\; n_{\text {e}} = 2.98$ and $n_{\text {o}} =2.43\; n_{\text {e}} = 3.88$ at the same wavelength, respectively. Assuming the mentioned values of the refractive indices, the birefringence $\Delta n$ dependent on the wavelength ($\lambda$) can be expressed as
The values of the ordinary and extraordinary refractive indices are estimated from Cauchy’s dispersion formula [40] written as
Table 1. The coefficients used in Cauchy’s dispersion formula for the estimation of the refractive indices of mercurous halides. The measured refractive indices exploited for coefficient estimation were obtained from [11].
For the polarizer construction, another key parameter is the transparency range. The transparency range for calomel covers a spectral band from $0.35 \;\mathrm {\mu }\textrm {m}$ up to $20\;\mathrm {\mu }\textrm {m}$. The range for kuzminite and moschelite is even wider, from $0.4 \;\mathrm {\mu }\textrm {m}$ up to $30 \;\mathrm {\mu }\textrm {m}$ and from $0.45 \;\mathrm {\mu }\textrm {m}$ to $40 \;\mathrm {\mu }\textrm {m}$, respectively [9]. This allows to exploit these materials in a highly broadband spectral region. However, design optimization of this wide-band polarizer would be significantly more complicated than for some specified narrow spectral bands. There are other parameters that are crucial for the exploitation of mercurous halides as materials for optical devices. These materials are very soft (hardness 1.5-2 according to the Mohs scale) and therefore problematic for polishing. Some other selected properties of mercurous halides, such as density or band gap, are summarized in Table 2.
Furthermore, due to the mentioned high birefringence (0.66, 0.86, 1.48 at 0.633 $\mathrm {\mu }\textrm {m}$), the transmission losses of radiation incident on a mercurous halide material are significant. The explanation is illustrated by Fig. 3. Let us assume radiation with wavelength 10.6 $\mathrm {\mu }\textrm {m}$ and the optic axis orientation of the mercurous halide crystal as in Fig. 3(a). The refractive index of the material, as well as the refractive angle and the transmittance, can be calculated based on the incidence angle $\theta _{\text {i}}$ on the surface of the air / mercurous halide (see Figs. 3(b)–(d). This simulated behaviour is based on standard Fresnel equations [6] and Snell’s law, incorporating the optical parameters of mercurous halides. All presented simulations of the mercurous halide characteristics may then be further exploited for the Wollaston modelling. Knowing the behaviour of the extraordinary refractive index in dependence on the incident angle (Fig. 3(b)), the parameters of both ordinary and extraordinary rays can be obtained. The refractive deviation of the beams can also be obtained 3(c)). The symmetrical shape of the curves in Fig. 3(c) also suggested all possible incident angle exploitation in the whole field of view. The symmetrical shape of the curves is also related to the symmetry of the crystalline materials. Moreover, focusing on transmittance maximization, Brewster’s angle for the ordinary beam around the incident angles $\pm 60^{\circ}$ for all materials at the presented wavelength can also be obtained, as well as the minimal transmittance difference at the normal incidence (Fig. 3(d)).
Fig. 3. Optical behaviour at the interface of air/mercurous halides (Fig. A). At this interface, the unpolarized incident radiation refracts as ordinary (S-polarized) and extraordinary (P-polarized) rays with different refractive indices and refraction angles. Fig. B shows the change in the refractive index $n$ against the incident angle of the radiation $\theta _{\text {i}}$. Figure C describes the angle of refraction $\theta _{\text {o,e}}$ versus the $\theta _{\text {i}}$. Fig. D shows the transmittance against the $\theta _{\text {i}}$. In all graphs, the straight lines correspond to the behaviour of the ordinary ray. Complementarily, the dashed lines correspond to the behaviour of the extraordinary ray. The symbols $n_\text {o}$ and $n_\text {e}$ represent the ordinary and extraordinary refractive index of mercurous halides.
In order to obtain the characterization of the Wollaston prism based on mercurous halides, the equations used for the above modelling are described and derived in more detail in the following section concerning the Wollaston prism characterization.
3. Wollaston prism parameters
In general, a Wollaston polarizer is constructed from two combined prisms. These two prisms are usually bonded together by suitable optical cement (immersion). Therefore, the transmitted radiation incident at the input facet of the polarizer undergoes several refractions and splits into two orthogonal highly polarized beams at the output of the polarizer separated by a particular angle. The standard scheme of the Wollaston polarizer with highlighted optical axes and cut angle of the wedge prisms is shown in Fig. 4.
Fig. 4. Wollaston scheme and illustration of unpolarized radiation propagation through it. The symbols L (length), W (width), and H (height) are the sizes of the Wollaston cube. $\alpha$ represents the cut angle of the wedge prisms, and $\delta$ corresponds to the output beam separation angle. $\theta _{\text {p}}$ is the refracted output angle of the P-polarized radiation and $\theta _{\text {s}}$ is the refracted output angle of the S-polarized radiation. The brown double arrow symbol indicates the orientation of the optic axis of each of the prisms.
The Wollaston polarizer can be evaluated according to several parameters. Many of these parameters, such as wavelength range or extinction ratio, depend on the properties and purity of the used optical crystalline material (substrate). Other parameters, such as clear aperture or field of view, are highly dependent on the size of the manufactured polarizer. Apart from the size and mentioned material properties, there are other key optimization parameters of the Wollaston polarizer that affect its polarizing behaviour, beam propagation, beam separation, and power transmission. Suppose ideally polished crystals with properly selected optical axes of each wedge prism. The two crucial parameters are the cut angle $\alpha$ of the wedge and the refractive index of the selected optical immersion $n_{\text {oc}}$ that bonds the wedges together. The optimization of the mentioned parameters for the Wollaston scheme based on mercurous halides is described in the following paragraphs.
Let us assume transmitted unpolarized radiation through the Wollaston polarizer. This radiation consists of two orthogonally polarized rays, annotated as the S-polarized beam and the P-polarized beam. The S-polarized beam corresponds to the beam perpendicular to the incident surface of the Wollaston prism facet. The P-polarized beam is a beam parallel to the incident surface of the Wollaston prism facet. The orientation of the optic axis of each wedge prism is crucial. For the first wedge of the Wollaston polarizer, the optic axis is standardly oriented in parallel with the plane of incidence and the input facet. The optic axis of the latter wedge is then perpendicular to the optic axis of the first one. This orientation decides which of the input polarizations corresponds to the ordinary or extraordinary ray transmitted through each wedge.
Since mercurous halides are optical materials with positive birefringence, the S-polarized ray transmits through the first wedge as an ordinary ray with an ordinary refractive index. Complementarily, the P-polarized ray behaves as an extraordinary ray with an extraordinary index of refraction. During transmission through the second wedge of the Wollaston polarizer, the roles of orthogonal polarizations exchange. Therefore, the S-polarized beam behaves as the extraordinary ray with the extraordinary refractive index, and the P-polarized beam is the ordinary ray with the ordinary index of refraction. The expected propagation of polarized radiation through the Wollaston polarizer is also indicated in Fig. 4. In general, the ordinary index of refraction is the same in all directions of propagation of the ordinary beam. However, the extraordinary refractive index depends on the crystal orientation and the angle of incidence of the extraordinary beam. Thus, the extraordinary index of refraction can be recovered from the expression
The values of the input and output refractive indices $n_{\text {in}}$ and $n_{\text {out}}$ in the above equations depend on whether the S- or P-polarized light propagates through the birefringent material as an ordinary or extraordinary wave. They also depend on the refractive index of the initial dielectric (usually air) and on the refractive index of a particular optical cement that bonds the polarizer wedges together. Moreover, the refraction and propagation of the S- and P-polarized radiation also depend on the selected crystal wedge cut angle $\alpha$. This angle significantly affects the refraction angles between the output and input interfaces of the first and second Wollaston wedge prisms. Suppose that $i$ is the index of the dielectric interfaces through which unpolarized radiation is transmitted. The overall transmittance [42] for each orthogonal polarization $T_{\text {s}}$ and $T_{\text {p}}$ can then be determined as
whereThe total reflected radiation for each orthogonal polarization $R_{\text {s}}$ and $R_{\text {p}}$ can be calculated from
whereFor the standard Wollaston polarizer, the number of dielectric interfaces that should be evaluated corresponds to $i = 1,\dots,4$. At the last interface $i = 4$ the important parameter for evaluating the Wollaston design is the separation angle $\delta$, which describes the output separation between the S- and P-polarized beams. This angle can be denoted as
where $\theta _{\text {s}}$ corresponds to the refracted output angle of the S-polarized radiation at the output surface of the Wollaston prism and $\theta _{\text {p}}$ is the refracted output angle of the P-polarized radiation at the output surface of the Wollaston prism. However, due to the dispersion of the refractive index in the transparency range of mercurous halides, key parameters such as transmittances $T_{\text {s}}$, $T_{\text {p}}$, and polarization separation output angle $\delta$ also change with the wavelength of input radiation.4. Parameter analysis and discussion
The previous section introduced the main maximization parameters of the Wollaston prism, such as the total energy transmission and the output beam separation angle. These parameters highly depend on the main optimization attributes, such as the prism wedge cut angle $\alpha$ and the optical cement refractive index $n_{\text {oc}}$ that have also been mentioned above. As mentioned, the overall construction sizes (length, width, height) of the Wollaston wedge prisms are also important but do not affect the total transmission and the output beam separation. In general, the process of growth and polishing of the mercurous halide crystals is complex and problematic. Currently, for the calomel prisms, the size of the crystal bowls is limited [24]. To optimize the Wollaston polarizer based on mercurous halides while taking into account the crystal size limits, a standard cube shape with sizes 10 $\text {mm}$, 10 $\text {mm}$, and 10 $\text {mm}$ for length, width, and height, respectively, has been selected. Because the transmittance and the output beam separation change with the input radiation wavelength, the optimization is performed for several selected wavelengths from the VIS to the TIR spectral band. These wavelengths are $0.633 \;\mathrm {\mu }\textrm {m}$, $0.946 \;\mathrm {\mu }\textrm {m}$, $1.152 \;\mathrm {\mu }\textrm {m}$, $0.208 \;\mathrm {\mu }\textrm {m}$, $3.391 \;\mathrm {\mu }\textrm {m}$, $5 \;\mathrm {\mu }\textrm {m}$, $7 \;\mathrm {\mu }\textrm {m}$, and $10.6 \;\mathrm {\mu }\textrm {m}$, and they are selected according to important laser wavelength lines in these bands. In addition, for most optimization processes, the normal incidence of the radiation at the input facet of the polarizer is considered.
In the first place, an optimal selection of the optical cement is necessary for the Wollaston construction. Due to the high birefringence, it is clear that for mercurous halides, the refractive index of the optical cement $n_{\text {oc}}$ should also be very high and, ideally, match the refractive index between the ordinary and extraordinary indices of the optical material. Subsequently, it should be compatible with the optical material in terms of its physical and chemical properties. However, there is no exact definition of how exactly the immersion should be selected.
In order to investigate the amount of unpolarized radiation transmission through the Wollaston polarizer at a particular wavelength (e.g. $10.6 \;\mathrm {\mu }\textrm {m}$ - $\mathrm {CO}_2$ laser), the transmission may be plotted against the refractive index of the optical cement and against the cut angle of the prism wedge (see Fig. 5). Currently, no crystal polarizers are available for this specific wavelength of ($10.6 \;\mathrm {\mu }\textrm {m}$) or, more precisely, for the TIR spectral band. Moreover, many applications could benefit from the possible utilization of polarizers operating in the TIR band [23,43–45]. Therefore, the transmission depending on the specific $\alpha$ and $n_{\text {oc}}$ is analyzed at the TIR wavelength in the following graphs.
Fig. 5. Transmittance of the Wollaston prism based on mercurous halides for the P-polarized beam $T_{\text {p}}$ and the S-polarized beam $T_{\text {p}}$ against the refractive index of the optical cement $n_{\text {oc}}$ and the cut angle $\alpha$ of the Wollaston prism. The wavelength $10.6 \;\mathrm {\mu }\textrm {m}$ has been selected for the presented analysis. Figures A, C, and E show the $T_{\text {p}}$ for calomel, kuzminite, and moschelite types of the Wollaston prism, respectively. Similarly, Figures B, D, and F show the $T_{\text {s}}$ for calomel, kuzminite and moschelite types of the Wollaston prism, respectively.
The results in Fig. 5 correspond to the calculated transmittance at the output of the polarizer for S- and P-polarized beams. These results show that only some cut angles of the wedged prism are acceptable, depending on the optical cement refractive index selection. Considering only an air gap between the Wollaston prism wedges, proper functionality of the polarizer is possible for prism cut angles up to $23^{\circ}$ for the calomel optical material and up to $22^{\circ}$, $19^{\circ}$ for kuzminite and moschelite, respectively. With an increase of the $n_{\text {oc}}$ between the Wollaston wedges, the possible prism cut angle increases. The increase of the possible $\alpha$ of the Wollaston prism also maximizes the possible transmittance of the P-polarized beam. Moreover, for a low $\alpha$, the increase of the $n_{\text {oc}}$ could mean an increase of the $T_{\text {p}}$ up to tens of percent. However, as shows Fig. 5, the increase of the $T_{\text {p}}$ is limited up to the specific $\alpha$. Calomel-based Wollaston prism with a refractive index of optical cement greater than 1.9 can be designed only up to a prism cut angle of 53$^{\circ}$. Similarly, the kuzminite-based Wollaston prism with a refractive index of optical cement greater than 2 can be designed up to the $\alpha$ 49$^{\circ}$ and the moschelite-based Wollaston polarizer with the $n_{\text {oc}}$ 2.2 can be designed up to the $\alpha$ equal to 43$^{\circ}$. Thus, for cut angles greater than those mentioned, the P-polarized beam experiences a total internal refraction at the output facet of the polarizer and the Wollaston polarizer does not operate properly at the exploited wavelength $10.6 \;\mathrm {\mu }\textrm {m}$.
The transmittance of the S-polarized beam through the Wollaston prism shows a different effect. With an increase in the cut angle $\alpha$, the S-polarized transmittance decreases. This can also be improved by increasing the refractive index of the optical cement $n_{\text {oc}}$. Similarly, as for the $T_{\text {p}}$ above, assuming a low rigid cut angle of the Wollaston prism, the $T_{\text {s}}$ can then be increased in the order of tens of percent by the increase of the $n_{\text {oc}}$. However, seeing the presented results, a trade-off must be considered between maximizing the transmission of S- or P- polarized beams for optimizing the Wollaston polarizer based on mercurous halides.
Focusing primarily on the P-polarized output ray and maximization of its transmittance, pairs of the $\alpha$ and the $n_{\text {oc}}$ for which the $T_{\text {p}}$ is at maximum can be selected. Complementarily, the maximal possible cut angle with the non-zero $T_{\text {p}}$ can also be selected. The results of combined $\alpha$ and $n_{\text {oc}}$ with max. $T_{\text {p}}$ and min. non-zero $T_{\text {p}}$ are shown in Fig. 6. Due to the change of the refractive index of mercurous halides, the optimal cut angle, which maximizes the P-polarized transmission, moves from lower angles to higher ones. Assuming the lowest and the highest wavelength equal to $0.633 \;\mathrm {\mu }\textrm {m}$ or $10.6 \;\mathrm {\mu }\textrm {m}$, the overall difference between the prism cut angle for a maximal transmission at these wavelengths is about $3^{\circ}$ for calomel, $3 ^{\circ}$ for kuzminite and $5 ^{\circ}$ for moschelite. Simultaneously, aiming for as broadband usage as possible, ideally from the VIS to the TIR spectral band, the ideal refractive index of optical material $n_{\text {oc}}$ should be approximately greater or equal to 2, depending on the mercurous halide material. As mentioned, for the $n_{\text {oc}}$ higher than 2, the increase of the $\alpha$ does not cause the expected increase of the $T_p$, due to the physical properties of mercurous halides and their refractive index. As both Fig. 5 and Fig. 6 illustrate, from the $n_{\text {oc}}$ around 2 a substantial increase of the $\alpha$ is not possible because the P-polarized beam is then totally reflected at the output facet of the Wollaston polarizer, and the polarizer does not work properly. Moreover, although the cut angles selected according to the IR wavelengths offer higher maximal values of the $T_{\text {p}}$ (see Fig. 5), the possible operational band is narrowing (see Fig. 6). Therefore, to ensure as broadband operation of the polarizer as possible, the Wollaston prism cut angle should be selected according to the lower VIS wavelengths rather than the IR ones.
Fig. 6. Figures A, C, and E illustrate the connected maximum transmittance peaks from Fig. 5 for each pair $n_{\text {oc}}$ (refractive index of optical cement) and $\alpha$ (cut angle of the Wollaston prism) at selected wavelengths. Figures B, D, and F show the connected maximal possible Wollaston prism cut angle $\alpha$ with the non-zero $T_\text {p}$ for mercurous halides.
Variations between the prism cut angle $\alpha$ and the refractive index of optical cement $n_{\text {oc}}$ can also be used to calculate the output beam separation angle $\delta$ of the Wollaston prism. The results of the output separation angle for the Wollaston prism based on mercurous halides are shown in Figs. 7(a), 7(c), and 7(e). These results are also calculated for an operational wavelength equal to $10.6 \;\mathrm {\mu }\textrm {m}$.
Fig. 7. The value of the output beam separation angle $\delta$ for the Wollaston prism based on mercurous halides against the refractive index of the optical cement $n_{\text {oc}}$ and the cut angle $\alpha$ of the Wollaston prism for the input radiation wavelength $10.6 \;\mathrm {\mu }\textrm {m}$. Figures A, C, and E show the $\delta$ for calomel, kuzminite, and moschelite types of the Wollaston prism, respectively. Figures B, D, and F show only $\delta$ versus $\alpha$ at selected wavelengths for calomel, kuzminite and moschelite types of the Wollaston prism, respectively ($n_{\text {oc}}$ = 1.5).
Apart from the same non-functional combinations of the $\alpha$ and the $n_{\text {oc}}$, as was shown above in the transmittance results (Fig. 5), it is apparent that the $\delta$ increases with the increasing $\alpha$. Nevertheless, the $\delta$ is almost independent on the increase of the $n_{\text {oc}}$. According to the presented results, the Wollaston design based on mercurous halides offers exceptional output beam separation angles because of their high refractive index. Assuming only an air-gap Wollaston design without any optical cement, the separation angle values between orthogonally polarized output beams can reach values up to $35^{\circ}$, $45^{\circ}$, and even $55^{\circ}$ for calomel, kuzminite, and moschelite, respectively. Suppose the $n_{\text {oc}}$ greater than 2 together with the $\alpha$ greater than $40^{\circ}$. The output separation angle $\delta$ can then reach values close to $70^{\circ}$, $80^{\circ}$, and even $110^{\circ}$ for calomel, kuzminite, and moschelite, respectively. For an operation in parts of the VIS spectral band, where mercurous halides are still transparent, the values of the beam separation angle may achieve even greater values than $90^{\circ}$ for calomel, $120^{\circ}$ for kuzminite, and $140^{\circ}$ for moschelite. Due to the independence of the $n_{\text {oc}}$ increase on the results of the $\delta$, the values of $\delta$ can be plotted only against the varying prism cut angle $\alpha$. This is shown for several wavelengths in Figs. 7(d) – 7(f). The dispersion of mercurous halides mainly causes the changes of the separation angle between the tested wavelengths. The higher the cut angle $\alpha$ of the prism wedge, the more significant the separation angle difference. For VIS and TIR wavelengths and prism cut angle about $30^{\circ}$, the difference may reach values greater than $10^{\circ}$, $15^{\circ}$, and $25^{\circ}$ for calomel, kuzminite, and moschelite, respectively.
In general, the problem of current adhesive-based optical cement is that its refractive index usually does not exceed the value of 1.6. According to the results presented above, this refractive index value is not optimal for the mercurous halide-based Wollaston polarizer. Moreover, the next issue may be a possible chemical reaction of mercurous halides with various substances. Assuming a currently available cyanoacrylate-based bonding [46] (compatible with mercurous halides) with a refractive index of approximately 1.5 across the VIS-TIR band, the graphs presented above (especially with respect to transmittance) can be simplified for optimal selection of the Wollaston prism wedge cut angle $\alpha$. The results of the Wollaston output transmittance depending on the prism cut angle $\alpha$ are shown in Fig. 8.
Fig. 8. Transmittance of the Wollaston prism based on mercurous halides for the P-polarized beam $T_{\text {p}}$ and the S-polarized beam $T_{\text {p}}$ against the cut angle $\alpha$ of the Wollaston prism at selected wavelengths. The refractive index of the optical cement $n_{\text {oc}}$ has been set equal to 1.5. Figures A, C, and E show the $T_{\text {p}}$ for calomel, kuzminite, and moschelite types of the Wollaston prism, respectively. Similarly, Figures B, D, and F show the $T_{\text {s}}$ for the calomel, kuzminite and moschelite types of the Wollaston prism, respectively.
Fig. 9. Zemax ray-racing of the Wollaston prism based on mercurous halides. The ray-racing is performed for reflections and refractions up to 5% of input energy. The ray colour is changed with every reflection or refraction. Input radiation is unpolarized with wavelength $0.633 \;\mathrm {\mu }\textrm {m}$. The normal incidence of the input radiation is used for the models. The dashed-black line represents the orientation of optic axis of the first prism. The left subfigure corresponds to the Wollaston prism based on the calomel optical material with the prism cut angle $\alpha _{\text {select}}$ equal to $31^{\circ}$. The middle subfigure corresponds to the Wollaston prism based on the kuzminite optical material with the prism cut angle $\alpha _{\text {select}}$ equal to $28^{\circ}$. The right subfigure corresponds to the Wollaston prism based on the moschelite optical material with the prism cut angle $\alpha _{\text {select}}$ equal to $21^{\circ}$.
The selection of a suitable cut angle for the Wollaston prism depends on the planned band of operation. As was mentioned, for as broadband operation as possible, the curve for a HeNe laser ($0.633 \; \mathrm {\mu }m$) from Fig. 8 can be used. Presented graphs show that the overall P-polarized transmittance does not exceed $0.8$ for calomel and is even lower for other mercurous halides. Therefore, exploiting an anti-reflex coating would be necessary to minimize losses at the interfaces. Regarding S-polarized transmittance, with an increase in the prism cut angle $\alpha$, the transmittance decreases, and thus it is necessary to determine how much energy loss is still acceptable. Using the results of Figs. 8(a), 8(c), 8(e), the suitable prism cut angle $\alpha _{\text {select}}$ has been selected by the following equations:
After determining the most critical fabrication parameters of a Wollaston prism from the above-presented analysis, the overall ray-tracing and precise modelling can be performed via a Zemax environment (see Fig. 9). The analysis proceeded in both SC (sequential) and NSC (non-sequential) modes of the Zemax environment. For the material description, data from Table 1 and Table 2 have been used. The sizes and parameters selected for the modelled polarizers based on mercurous halides are summarized in Table 3 together with calculated features such as transmittance $T_{\text {p}}$, $T_{\text {s}}$, and output beam separation angle $\delta$. For more illustrative clarity, only one unpolarized direct beam with wavelength $0.633 \; \mathrm {\mu }\textrm {m}$ approaching the center of the input polarizer interface has been selected. The ray-tracing process has been monitored for an input power level of the radiation beam equal to 1 W. The rays within the crystal have been tracked up to a power level equal to 0.05 W, up to 100 intersections, or up to 500 segments per ray. According to the presented models, the ideally polished Wollaston prism based on mercurous halides with defined cut angles $\alpha$ should split the unpolarized input beam exactly into two orthogonal polarizations, as expected from the above calculations. The results calculated above are also confirmed by the polarization output pupil graphs shown in Fig. 10. The overall transmittance of P-polarized radiation through the Wollaston prism is close to (considering the interpolations effects of Zemax) the expected 0.73 for calomel and 0.66 or 0.52 for kuzminite and moschelite, respectively. The S-polarized output pupil modelling confirms the same results, where the transmission is close to the expected values of 0.58, 0.5 and 0.33 for calomel, kuzminite, and moschelite, respectively. Significant output beam separation angles are also shown for several wavelengths from $0.633 \; \mathrm {\mu }\textrm {m}$ up to $10.6 \; \mathrm {\mu }\textrm {m}$ in the footprint diagrams in Fig. 10. Separation angles are for the center of the footprints close to the expected values $44 ^{\circ}$ for calomel, $50 ^{\circ}$ for kuzminite, and $70 ^{\circ}$ for moschelite.
Fig. 10. Polarization pupil maps and footprint diagrams of output polarized beams obtained from the Zemax-modeled Wollaston prisms based on mercurous halides. The first row corresponds to the results of the output polarization pupil of the P-polarized beam. The second row corresponds to the output polarization pupil of the S-polarized beam. The last row corresponds to the output footprint diagram of both polarizations coming out from the Wollaston polarizer. The results of the calomel-based, kuzminite-based, and moschelite-based Wollaston prisms are in the first, second, and third columns, respectively.
Table 3. Parameters of Zemax-modeled Wollaston polarizers based on mercurous halides
The results of the presented analysis of mercurous halide-based Wollaston polarizer can be compared to the parameters of conventional commercial Wollaston polarizers. The commercial Wollaston polarizes based on the quartz ($\mathrm {SiO}_2$) substrate offer a limited wavelength range from $0.2 \;\mathrm {\mu }\textrm {m}$ to $2.3 \;\mathrm {\mu }\textrm {m}$ with the output separation angle in order of degrees ($2^{\circ} - 3 ^{\circ}$). Other commercial polarizers based on calcite ($\mathrm {CaCO}_3$), magnesium fluoride ($\mathrm {MgF}_2$), yttrium orthovanadate ($\mathrm {YVO}_4$), and alpha-barium borate ($\alpha \mathrm {Ba}\mathrm {B}_2\mathrm {O}_4$) offer the wavelength range $0.35 \;\mathrm {\mu }\textrm {m} - 2.3 \;\mathrm {\mu }\textrm {m}$, $0.2 \;\mathrm {\mu }\textrm {m} - 6 \;\mathrm {\mu }\textrm {m}$, $0.4 \;\mathrm {\mu }\textrm {m} - 4 \;\mathrm {\mu }\textrm {m}$, and $0.19 \;\mathrm {\mu }\textrm {m} - 3.5 \;\mathrm {\mu }\textrm {m}$, respectively. The output separation angles of the Wollaston polarizers based on these materials are ($16.7^{\circ} - 22.5 ^{\circ}$), ($1.7 ^{\circ} - 3.1 ^{\circ}$), ($19.6^{\circ} - 22.3 ^{\circ}$), and ($15^{\circ} - 27 ^{\circ}$), respectively. Thus, in terms of wavelength range and output separation angle, the mercurous halide-based Wollaston polarizer offers superior parameters compared to the remaining commercial Wollaston polarizers.
Assessing the presented Zemax models of mercurous halide-based Wollaston, it is clear that the fabricated Wollaston prism with the correct orientation of the optic axes and polishing should be functional as expected for all VIS-TIR wavelengths. Currently, the limiting factor of the polarizer construction is the complicated growth and fabrication of mercurous halide crystals. Some of the recent advances in this field appear to be promising. Currently, calomel crystals are grown to a large enough size and some tests are carried out with them [18,24]. Similarly, there are some advances in kuzminite growth [25,27–30]. Still, the mentioned research appears only as the first stage of real calomel and kuzminite exploitation as photonic devices. Moreover, the growth of moschelite crystals appears to be the most complicated because there is not any recently published research on this topic. Nevertheless, the presented study mainly aimed to show the exceptional properties of mercurous halides and their possible exploitation as polarizing devices. After the point of sufficient size and quality of the mercurous halide crystals is reached, the construction of a Wollaston polarizer based on these materials will extend several photonic applications in an infrared spectral band, such as microscopy or hyperspectral imaging.
5. Conclusion
This paper provided analysis and proposed construction parameters of a Wollaston prism based on the mercurous halide ($\mathrm {Hg}_2\mathrm {X}_2$) crystals. Due to the exceptional properties of mercurous halides such as calomel ($\mathrm {Hg}_2\mathrm {Cl}_2$), kuzminite ($\mathrm {Hg}_2\mathrm {Br}_2$), and moschelite ($\mathrm {Hg}_2\mathrm {I}_2$), the Wollaston prism based on these materials can operate within a wide spectral range, from VIS to TIR spectral bands. In addition, due to the high birefringence, the Wollaston prism based on mercurous halides offers a high separation angle of the output orthogonally polarized beams. The main Wollaston parameters, such as the transmittance of the polarized beams and their output separation angle, affect two main factors, the cut angle of the Wollaston wedge prism and the refractive index of the used optical cement that bonds them together. This paper provided an analysis on the determination of these factors and the subsequent selection of their optimal values. While assuming a standard refractive index of adhesive bonding used as an optical cement equal to 1.5, the optimal selection of a prism cut angle of a Wollaston polarizer has been determined equal to $31^{\circ}$ for calomel, $28^{\circ}$ kuzminite, and $21^{\circ}$ for moschelite. On the basis of these optimized factors, precise modelling of a Wollaston prism based on mercurous halides has been performed using a Zemax optic studio. The functional operation of such polarizer devices has been confirmed. Thus, the Wollaston polarizer based on mercurous halides may find many applications in photonics setups and devices and additionally also extend the fields of microscopy and hyperspectral imaging in several infrared spectral bands.
Funding
Grant Agency of the Czech Technical University in Prague (SGS20/179/OHK3/3T/13).
Disclosures
“The authors declare no conflicts of interest.”
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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